Electromagnetic Induction - College Physics

1. According to Lenz's law, when the magnetic flux through a conducting loop increases, the direction of the induced current in the loop will be such that:
   Answer: the induced magnetic field opposes the original magnetic field that caused the flux change When you encounter questions about electromagnetic induction, you're dealing with one of physics's most elegant principles: Lenz's law, which governs how nature responds to changing magnetic flux through conducting loops. Lenz's law states that induced currents always flow in a direction that creates a magnetic field opposing the change that produced them. This is nature's way of "resisting" changes in magnetic flux. When magnetic flux through a loop increases, the induced current generates its own magnetic field that points opposite to the original field, effectively trying to reduce the total flux back toward its original value. Answer B correctly captures this opposition principle. The induced magnetic field always works against the change, not with it. Answer A represents the opposite misconception - that induced effects reinforce the original change. If this were true, any small flux change would create a runaway positive feedback loop, violating energy conservation. Answer C incorrectly suggests the induced current direction depends on external circuits. Lenz's law is an intrinsic property of electromagnetic induction that doesn't require any external reference. Answer D confuses Lenz's law with power considerations. While induced currents do dissipate power through resistance, Lenz's law specifically governs current direction based on flux opposition, not power maximization. Remember this key pattern: Lenz's law is always about opposition to change. Whether flux is increasing or decreasing, the induced current creates a magnetic field that "fights back" against whatever is changing. This opposition principle appears frequently in electromagnetic induction problems.
2. An airplane with a wingspan of 40 m flies horizontally at 250 m/s through Earth's magnetic field. If the vertical component of Earth's field is $5.0 \times 10^{-5}$ T, what is the potential difference induced between the wing tips?
   Answer: 0.50 V This question tests electromagnetic induction, specifically motional EMF - the voltage created when a conductor moves through a magnetic field. When you see a moving conductor in a magnetic field, think about Faraday's law and the relationship between motion, field strength, and induced voltage. The motional EMF formula is $\varepsilon = BLv$, where B is the magnetic field strength, L is the length of the conductor (wingspan), and v is the velocity perpendicular to the field. Here, the airplane's 40 m wingspan acts as a conductor moving horizontally through Earth's vertical magnetic field. Substituting the values: $\varepsilon = (5.0 \times 10^{-5} \text{ T})(40 \text{ m})(250 \text{ m/s}) = 0.50 \text{ V}$ Choice A (0.25 V) likely comes from using half the wingspan (20 m instead of 40 m) - a common error where students might think only one wing tip matters. Choice C (1.0 V) could result from doubling the correct answer or using an incorrect magnetic field value. Choice D (2.0 V) represents a more significant calculation error, possibly from incorrectly handling the scientific notation or confusing the magnetic field components. The correct answer is B (0.50 V). Remember that motional EMF problems always require three key components: the magnetic field strength, the length of the conductor perpendicular to the field, and the velocity perpendicular to the field. Make sure you're using the full wingspan as your conductor length, not just the distance from center to tip.
3. An inductor with self-inductance $L = 0.50$ H carries a current that decreases linearly from 4.0 A to 1.0 A over a time period of 2.0 s. What is the magnitude of the self-induced EMF during this time interval?
   Answer: 0.75 V When you encounter problems involving changing currents in inductors, you're dealing with Faraday's law of electromagnetic induction. An inductor opposes changes in current by generating a self-induced EMF (electromotive force) that's proportional to how quickly the current changes. The self-induced EMF in an inductor is given by $\varepsilon = -L \frac{dI}{dt}$, where $L$ is the self-inductance and $\frac{dI}{dt}$ is the rate of current change. Since the current decreases linearly from 4.0 A to 1.0 A over 2.0 s, the rate of change is constant: $\frac{dI}{dt} = \frac{1.0 - 4.0}{2.0} = -1.5$ A/s. Substituting into the formula: $\varepsilon = -0.50 \text{ H} \times (-1.5 \text{ A/s}) = 0.75$ V. The magnitude is 0.75 V, making B correct. Looking at the wrong answers: A (0.25 V) results from incorrectly using only the final current value divided by time (1.0/2.0 × 0.50). C (1.25 V) comes from using the initial current instead of the current change (4.0/2.0 × 0.50 - 0.75). D (1.50 V) is what you'd get if you forgot to include the inductance value and just used the current change rate. Remember this key pattern: for linear current changes in inductors, always calculate $\frac{\Delta I}{\Delta t}$ first, then multiply by the inductance. The negative sign in Faraday's law just indicates direction—problems asking for magnitude want the absolute value.
4. A circular conducting loop of radius $r = 0.15 \text{ m}$ is located in a magnetic field that varies according to $B(t) = 0.20\cos(120\pi t)$ T, where the field is uniform and perpendicular to the loop. At what time $t > 0$ does the magnitude of the induced EMF first reach its maximum value?
   Answer: $t = \frac{1}{240} \text{ s}$ The flux is $\Phi_B = B(t) \cdot \pi r^2 = 0.20\cos(120\pi t) \cdot \pi r^2$. The induced EMF is $\mathcal{E} = -\frac{d\Phi_B}{dt} = 0.20 \cdot 120\pi \sin(120\pi t) \cdot \pi r^2$. The magnitude $|\mathcal{E}|$ is maximum when $|\sin(120\pi t)| = 1$, which first occurs when $120\pi t = \frac{\pi}{2}$, giving $t = \frac{1}{240} \text{ s}$.
5. A solenoid with 200 turns and cross-sectional area $2.0 \times 10^{-3} \text{ m}^2$ is placed in a magnetic field. If the magnetic field through the solenoid changes from 0.10 T to 0.40 T in 0.50 s, what is the magnitude of the induced EMF?
   Answer: $2.4 \times 10^{-1}$ V When you encounter a problem about changing magnetic fields and solenoids, you're dealing with electromagnetic induction and Faraday's Law. The key insight is that a changing magnetic flux through a coil induces an EMF proportional to both the rate of change and the number of turns. Faraday's Law states that the induced EMF equals $|\varepsilon| = N \frac{\Delta \Phi}{\Delta t}$, where N is the number of turns and $\Delta \Phi$ is the change in magnetic flux. Since magnetic flux is $\Phi = BA$ for a uniform field perpendicular to the area, the change in flux is $\Delta \Phi = A \Delta B$. Let's calculate: $\Delta B = 0.40 - 0.10 = 0.30$ T, so $\Delta \Phi = (2.0 \times 10^{-3})(0.30) = 6.0 \times 10^{-4}$ Wb. Therefore: $|\varepsilon| = 200 \times \frac{6.0 \times 10^{-4}}{0.50} = 200 \times 1.2 \times 10^{-3} = 2.4 \times 10^{-1}$ V. Answer A ($2.4 \times 10^{-2}$ V) results from forgetting to multiply by the number of turns—using only the flux change rate. Answer B ($1.2 \times 10^{-1}$ V) comes from correctly calculating $\frac{\Delta \Phi}{\Delta t}$ but forgetting the factor of 200 turns, then making an error. Answer D ($4.8 \times 10^{-1}$ V) likely results from incorrectly doubling the final answer or making an arithmetic error. Remember: in induction problems, always account for the number of turns in coils or solenoids—it's a common source of errors on exams.
6. A metal disk rotates about its center in a uniform magnetic field that is parallel to the rotation axis. As the disk rotates, what happens regarding electromagnetic induction?
   Answer: An EMF is induced between the center and rim of the disk due to the motion of conducting material When analyzing electromagnetic induction problems involving rotating conductors, you need to consider how the motion of charge carriers creates separation and potential differences, even when the overall magnetic flux remains unchanged. As the metal disk rotates in the uniform magnetic field, free electrons within the conducting material experience a magnetic force. Using the right-hand rule, electrons moving with the disk's rotation feel a force directed radially (either inward or outward, depending on the field direction). This force causes charge separation: electrons accumulate preferentially at either the center or rim, leaving positive charge at the opposite location. This charge separation creates an electric field and corresponding EMF between the center and rim, making choice A correct. Choice B incorrectly focuses only on magnetic flux through the disk's area. While the total flux does remain constant, EMF can arise from charge motion within the conductor itself, not just from changing flux linkage. Choice C represents a common misconception that EMF requires a time-varying magnetic field. However, motional EMF ($\mathcal{E} = Blv$) occurs whenever conductors move through static fields. Choice D misunderstands the mechanism—while rim velocity is indeed maximum, the EMF develops across the entire radius due to the radial force on all moving charges, not just at the edges. Remember: electromagnetic induction has two main causes—changing magnetic flux (Faraday's law) and conductor motion in magnetic fields (motional EMF). Always consider both possibilities when analyzing induction problems.
7. A transformer has a primary coil with 100 turns and a secondary coil with 500 turns. If an alternating voltage of 120 V is applied to the primary coil, what is the voltage across the secondary coil, assuming the transformer is ideal?
   Answer: 600 V When you encounter transformer problems, you're dealing with electromagnetic induction and the relationship between coil turns and voltage. The key principle is that an ideal transformer changes voltage in proportion to the ratio of turns between its coils. For an ideal transformer, the voltage relationship follows: $\frac{V_s}{V_p} = \frac{N_s}{N_p}$, where $V_s$ and $V_p$ are secondary and primary voltages, and $N_s$ and $N_p$ are the number of turns in each coil. Given: $N_p = 100$ turns, $N_s = 500$ turns, and $V_p = 120$ V. Solving for the secondary voltage: $V_s = V_p \times \frac{N_s}{N_p} = 120 \text{ V} \times \frac{500}{100} = 120 \times 5 = 600 \text{ V}$ This confirms answer D is correct. Looking at the wrong answers: A) 24 V represents using the inverse ratio (100/500 = 0.2, so 120 × 0.2 = 24), which would apply if you confused which coil was primary versus secondary. B) 120 V suggests no voltage change occurred, ignoring the transformer's turn ratio entirely. C) 240 V might result from incorrectly using a 2:1 ratio instead of the actual 5:1 ratio, perhaps from misreading the given values. Remember this pattern: transformers are "step-up" when the secondary has more turns (voltage increases) and "step-down" when it has fewer turns (voltage decreases). Always check whether your calculated voltage makes sense given the turn ratio direction.
8. A circular loop of wire with radius 0.10 m is placed in a uniform magnetic field of 0.50 T pointing perpendicular to the plane of the loop. If the magnetic field decreases uniformly to zero over a time interval of 2.0 s, what is the magnitude of the induced EMF in the loop?
   Answer: $7.9 \times 10^{-3}$ V When you encounter problems involving changing magnetic fields and loops of wire, you're dealing with electromagnetic induction and Faraday's law. This fundamental principle states that a changing magnetic flux through a loop induces an EMF proportional to the rate of change. To find the induced EMF, use Faraday's law: $|\varepsilon| = \left|\frac{d\Phi_B}{dt}\right|$, where $\Phi_B = BA$ is the magnetic flux. Since the field is uniform and perpendicular to the loop, the flux is simply $\Phi_B = BA = B\pi r^2$. The magnetic field changes from 0.50 T to 0 T over 2.0 s, so $\frac{dB}{dt} = \frac{0 - 0.50}{2.0} = -0.25$ T/s. The area of the circular loop is $\pi r^2 = \pi(0.10)^2 = 0.0314$ m². Therefore: $|\varepsilon| = \left|\frac{d\Phi_B}{dt}\right| = \left|\frac{dB}{dt}\right| \times A = 0.25 \times 0.0314 = 7.9 \times 10^{-3}$ V. This confirms answer B is correct. Answer A ($3.9 \times 10^{-3}$ V) likely results from using the wrong area calculation or incorrectly halving the rate of change. Answer C ($1.6 \times 10^{-2}$ V) doubles the correct value, possibly from a sign error or miscalculating the time derivative. Answer D ($2.5 \times 10^{-2}$ V) suggests using diameter instead of radius in the area calculation. Remember: in Faraday's law problems, carefully track what's changing with time and always use the full circular area $\pi r^2$, not just the radius or diameter.
9. A conducting loop with area 0.050 m² is oriented at a 60° angle to a uniform magnetic field of 0.30 T. If the loop is rotated so that it becomes perpendicular to the field in 0.25 s, what is the average induced EMF during this rotation?
   Answer: 30 mV This problem tests Faraday's law of electromagnetic induction, which relates changing magnetic flux to induced EMF. When you see a rotating loop in a magnetic field, think about how the flux changes as the orientation changes. The key is calculating the change in magnetic flux. Magnetic flux is given by $\Phi = BA\cos\theta$, where B is the magnetic field strength, A is the loop area, and θ is the angle between the field and the normal to the loop. Initially, the loop is at 60° to the field, so $\Phi_i = (0.30)(0.050)\cos(60°) = 0.015 \times 0.5 = 0.0075$ Wb. When perpendicular to the field, θ = 0°, so $\Phi_f = (0.30)(0.050)\cos(0°) = 0.015 \times 1 = 0.015$ Wb. The change in flux is $\Delta\Phi = 0.015 - 0.0075 = 0.0075$ Wb. Using Faraday's law, the average induced EMF is $|\varepsilon| = \frac{\Delta\Phi}{\Delta t} = \frac{0.0075}{0.25} = 0.030$ V = 30 mV. Answer A (7.5 mV) likely comes from using only the initial flux divided by time, ignoring the final state. Answer B (15 mV) might result from using the final flux alone instead of the change. Answer C (22.5 mV) could come from incorrectly calculating the cosine values or making an arithmetic error in the flux difference. Remember: Faraday's law depends on the change in flux, not the flux itself. Always calculate both initial and final flux values, then find their difference.
10. A conducting bar of length 0.80 m rotates about one end in a uniform magnetic field of 0.15 T perpendicular to the plane of rotation. If the bar completes 2.0 revolutions per second, what is the EMF induced between the ends of the bar?
    Answer: 0.60 V When you see a conducting bar rotating in a magnetic field, you're dealing with motional EMF caused by electromagnetic induction. The key insight is that different parts of the bar move at different speeds, creating a varying EMF along its length. For a bar rotating about one end, you need to integrate the EMF contributions from each infinitesimal segment. Each small element at distance $r$ from the pivot moves with velocity $v = \omega r$, where $\omega$ is the angular velocity. The EMF for each element is $dEMF = Bv \, dr = B\omega r \, dr$. First, convert the rotation rate: $\omega = 2.0 \text{ rev/s} \times 2\pi = 4\pi \text{ rad/s}$. Integrating along the bar's length: $EMF = \int_0^L B\omega r \, dr = B\omega \int_0^L r \, dr = B\omega \frac{L^2}{2}$. Substituting values: $EMF = (0.15)(4\pi)\frac{(0.80)^2}{2} = (0.15)(4\pi)(0.32) = 0.60\pi \approx 0.60 \text{ V}$. Choice A (0.30 V) likely comes from forgetting the factor of $\frac{1}{2}$ in the integration, or using $L$ instead of $L^2$. Choice C (0.96 V) might result from calculation errors or incorrectly handling the units. Choice D (1.2 V) could come from omitting the $\frac{1}{2}$ factor and making other computational mistakes. Study tip: For rotating bars, always remember that EMF depends on $L^2$ (length squared), not just $L$. The $\frac{1}{2}$ factor comes from integration and is easy to forget under time pressure.
11. A rectangular conducting loop is pulled out of a magnetic field region at constant velocity. While the loop is exiting the field, the power dissipated in the loop's resistance is 2.0 W. If the velocity is doubled while keeping all other parameters the same, what will be the new power dissipated?
    Answer: 8.0 W When a conducting loop moves through a magnetic field, you're dealing with electromagnetic induction and Faraday's law. The key insight is understanding how induced EMF relates to velocity, and how power depends on that EMF. As the loop exits the magnetic field at velocity $v$, the changing magnetic flux induces an EMF given by $\varepsilon = Blv$, where $B$ is the magnetic field strength and $l$ is the width of the loop in the field. The power dissipated in the loop's resistance is $P = \frac{\varepsilon^2}{R} = \frac{(Blv)^2}{R}$. Notice that power is proportional to the square of velocity: $P \propto v^2$. When you double the velocity, the induced EMF doubles, but since power depends on EMF squared, the power increases by a factor of four. Starting from 2.0 W, doubling the velocity gives $P_{new} = 4 \times 2.0 = 8.0$ W. Looking at the wrong answers: A) 1.0 W suggests power is inversely proportional to velocity, which contradicts the physics. B) 2.0 W implies power doesn't change with velocity, ignoring the velocity dependence entirely. C) 4.0 W represents the correct factor of four increase but might come from incorrectly thinking power is directly proportional to velocity rather than velocity squared. Remember this pattern: in electromagnetic induction problems, EMF is linear in velocity, but power (which depends on EMF squared) scales as velocity squared. Watch for this $v^2$ relationship whenever power dissipation is involved in moving conductor problems.
12. A conducting loop is pulled away from a long straight wire carrying a steady current $I$. As the loop moves away with constant velocity, the direction of the induced current in the loop (as viewed from above) will be:
    Answer: Clockwise, because the flux through the loop is decreasing When you encounter electromagnetic induction problems involving moving loops and current-carrying wires, apply Lenz's law systematically: the induced current always opposes the change causing it. Here's the step-by-step analysis: As the loop moves away from the wire, the magnetic field through the loop weakens because magnetic field strength decreases with distance. Since the wire's current creates a magnetic field that points into the page on one side and out of the page on the other (use the right-hand rule), the magnetic flux through the loop is decreasing. By Lenz's law, the induced current must create a magnetic field to oppose this decrease—meaning it tries to maintain the original flux direction. Using the right-hand rule again, a clockwise current in the loop will generate a magnetic field in the same direction as the original field from the wire, opposing the flux decrease. Let's examine why the other options fail: Choice A is too vague—current direction in the wire doesn't directly determine the loop's induced current direction. Choice B incorrectly states the direction; while the flux is decreasing, the induced current is clockwise, not counterclockwise. Choice C mentions the correct direction but gives the wrong reasoning—it's not simply because the loop moves away, but specifically because moving away causes decreasing flux. Remember this pattern: when magnetic flux through a loop decreases, the induced current flows in whatever direction creates a magnetic field that would restore the original flux. Always combine Faraday's law with Lenz's law for complete analysis.
13. A solenoid with $N = 500$ turns, length $\ell = 0.20 \text{ m}$, and cross-sectional area $A = 0.0040 \text{ m}^2$ carries a current that changes according to $I(t) = 3.0\sin(60t)$ A. What is the magnitude of the self-induced EMF when the current is at its maximum value?
    Answer: Zero The self-induced EMF is $\mathcal{E} = -L\frac{dI}{dt}$. Since $I(t) = 3.0\sin(60t)$, we have $\frac{dI}{dt} = 3.0 \times 60\cos(60t) = 180\cos(60t)$. The current is maximum when $\sin(60t) = \pm 1$, which occurs when $\cos(60t) = 0$. Therefore, $\frac{dI}{dt} = 0$ when the current is maximum, making the self-induced EMF zero.
14. A rectangular loop moves with constant velocity into a region where there are two uniform magnetic fields: the first half of the region has field $B_1 = 0.20$ T and the second half has field $B_2 = 0.50$ T, both pointing in the same direction. As the loop moves from the first region into the second region, what happens to the induced EMF?
    Answer: The EMF increases because the magnetic field strength increases from $B_1$ to $B_2$ When you encounter problems about loops moving between different magnetic field regions, focus on Faraday's law: the induced EMF depends on the rate of change of magnetic flux through the loop. As the rectangular loop transitions from the first region ($B_1 = 0.20$ T) into the second region ($B_2 = 0.50$ T), the magnetic flux through the loop changes. During this transition, part of the loop experiences $B_1$ while the other part experiences $B_2$. The rate of flux change is $\frac{d\Phi}{dt} = (B_2 - B_1) \cdot v \cdot w$, where $v$ is the loop's velocity and $w$ is its width. Since $B_2 > B_1$, the flux increases as more of the loop enters the stronger field region. Choice A correctly identifies that the EMF increases because the magnetic field strength increases. The induced EMF is proportional to the field difference $(B_2 - B_1)$, so a larger $B_2$ means greater EMF. Choice B is wrong because the EMF doesn't decrease—it increases due to the stronger field. Choice C incorrectly assumes constant velocity means constant EMF, but EMF depends on flux change rate, not just velocity. The field difference is what matters here. Choice D is incorrect because while the EMF magnitude does depend on the field difference, it doesn't change sign during this transition—the flux is continuously increasing. Remember: in electromagnetic induction problems, always consider what's changing. Constant velocity doesn't guarantee constant EMF if the magnetic environment changes. Focus on flux change rates, not just motion speeds.
15. A square loop with side length 0.20 m and resistance 5.0 Ω is placed perpendicular to a magnetic field that varies according to $B(t) = 0.10t^2$ T, where $t$ is in seconds. What is the magnitude of the induced current at $t = 3.0$ s?
    Answer: 4.8 mA This problem tests electromagnetic induction using Faraday's law, which states that a changing magnetic flux through a loop induces an EMF (and therefore current) proportional to the rate of flux change. To find the induced current, you need to apply Faraday's law: $|\varepsilon| = \left|\frac{d\Phi_B}{dt}\right|$, where the magnetic flux is $\Phi_B = B \cdot A$. Since the loop is perpendicular to the field, $\Phi_B = B(t) \cdot A = 0.10t^2 \cdot (0.20)^2 = 0.004t^2$ Wb. Taking the time derivative: $\frac{d\Phi_B}{dt} = \frac{d}{dt}(0.004t^2) = 0.008t$ V. At $t = 3.0$ s: $|\varepsilon| = 0.008 \times 3.0 = 0.024$ V. Using Ohm's law: $I = \frac{\varepsilon}{R} = \frac{0.024}{5.0} = 0.0048$ A = 4.8 mA. Answer A (4.8 mA) is correct. Answer B (9.6 mA) likely comes from forgetting to square the side length when calculating area, using $A = 0.20$ instead of $A = 0.04$ m². Answer C (2.4 mA) might result from incorrectly taking the derivative, getting $0.004t$ instead of $0.008t$. Answer D (1.2 mA) could come from multiple errors, such as using the wrong time value or misapplying the derivative. Remember: Faraday's law problems always require taking a time derivative of the flux. Practice identifying what's changing with time (here, the magnetic field strength), then carefully apply the chain rule when differentiating.
16. A metal rod of length 0.30 m moves with constant velocity 5.0 m/s perpendicular to a uniform magnetic field of 0.20 T. The rod, field, and velocity are all mutually perpendicular. What is the magnitude of the motional EMF induced across the rod?
    Answer: 0.30 V When you encounter a moving conductor in a magnetic field, you're dealing with motional EMF - the voltage induced when a conductor cuts through magnetic field lines. This is a direct application of Faraday's law of electromagnetic induction. For a rod moving perpendicular to a uniform magnetic field, the motional EMF is given by $\varepsilon = BLv$, where B is the magnetic field strength, L is the length of the conductor, and v is the velocity. Since the rod, field, and velocity are mutually perpendicular (the ideal geometry), you can use this formula directly. Substituting the given values: $\varepsilon = (0.20 \text{ T})(0.30 \text{ m})(5.0 \text{ m/s}) = 0.30 \text{ V}$. This confirms answer B is correct. Let's examine why the other options are wrong. Choice A (0.15 V) would result if you mistakenly used half the velocity (2.5 m/s instead of 5.0 m/s) or half the magnetic field strength. Choice C (0.60 V) occurs if you double one of the parameters - perhaps confusing the rod length or using 10 m/s for velocity. Choice D (1.0 V) suggests using incorrect values like 1.0 T for the magnetic field instead of 0.20 T. Remember this key pattern: motional EMF problems involving perpendicular motion always use $\varepsilon = BLv$. When the geometry isn't perfectly perpendicular, you'll need the more general form involving sine or cosine of angles. Always check that your units work out to volts.
17. Refer to the diagram. A conducting bar slides along two parallel conducting rails in a uniform magnetic field $\vec{B}$ pointing out of the page. The bar moves to the right with velocity $\vec{v}$. What is the direction of the induced current in the bar?
    Answer: From top to bottom in the bar, creating a clockwise current in the circuit loop As the bar moves right, the area of the loop increases, so flux ($\Phi = BA$) through the loop increases. By Lenz's law, the induced current opposes this increase by creating a magnetic field into the page. Using the right-hand rule, this requires a clockwise current, meaning current flows from top to bottom in the moving bar. Choice B gives the wrong direction. Choice C ignores the changing flux. Choice D incorrectly suggests alternating current. Choice E misunderstands current flow direction in conductors.
18. A conducting loop with resistance $R = 2.0 \text{ Ω}$ is placed in a uniform magnetic field $\vec{B} = 0.40 \text{ T}$ pointing perpendicular to the loop. The area of the loop changes according to $A(t) = 0.20 - 0.050t^2$ (in $\text{m}^2$). What is the magnitude of the induced current at $t = 2.0 \text{ s}$?
    Answer: $0.080 \text{ A}$ Using Faraday's law: $|\mathcal{E}| = \left|\frac{d\Phi_B}{dt}\right| = \left|B\frac{dA}{dt}\right|$. First find $\frac{dA}{dt} = \frac{d}{dt}(0.20 - 0.050t^2) = -0.10t$. At $t = 2.0 \text{ s}$: $\frac{dA}{dt} = -0.20 \text{ m}^2\text{/s}$. Therefore $|\mathcal{E}| = |0.40 \times (-0.20)| = 0.080 \text{ V}$. The induced current is $I = \frac{|\mathcal{E}|}{R} = \frac{0.080}{2.0} = 0.080 \text{ A}$.
19. Two identical circular loops are placed parallel to each other. When the current in the first loop increases, what happens to the second loop according to electromagnetic induction principles?
    Answer: A current is induced in the second loop in the opposite direction to the current in the first loop When you encounter problems involving changing magnetic fields and loops of wire, you're dealing with Faraday's law of electromagnetic induction and Lenz's law. These principles govern how changing magnetic flux through a conductor induces an electric current. As current increases in the first loop, it creates a stronger magnetic field that passes through the second loop. This changing magnetic flux induces an electromotive force (EMF) in the second loop according to Faraday's law: $\varepsilon = -\frac{d\Phi_B}{dt}$. The induced EMF drives a current through the second loop. Lenz's law determines the direction of this induced current: it flows in whatever direction opposes the change causing it. Since the magnetic field from the first loop is increasing through the second loop, the induced current in the second loop creates its own magnetic field that opposes this increase. This requires the induced current to flow in the opposite direction to the current in the first loop. Option A is incorrect because it violates Lenz's law—currents in the same direction would reinforce rather than oppose the change. Option C reflects a common misconception that physical connection is required for electromagnetic induction; magnetic fields can induce currents across empty space. Option D is wrong because the material properties don't determine whether induction occurs (though they affect the magnitude of induced current). Remember this key pattern: in electromagnetic induction problems, nature always opposes change. When magnetic flux through a loop increases, the induced current creates a field that tries to reduce that flux.
20. A conducting loop is moved at constant speed from a region with no magnetic field into a region with uniform magnetic field $\vec{B}$. During the time the loop is entering the field region, which statement best describes the induced current?
    Answer: Current flows in a direction such that its magnetic field opposes the external field $\vec{B}$ When you encounter problems involving moving conductors and magnetic fields, you're dealing with electromagnetic induction governed by Faraday's law and Lenz's law. The key insight is that changing magnetic flux through a loop induces an EMF and current. As the conducting loop enters the magnetic field region, the magnetic flux through the loop increases continuously. This changing flux induces an EMF according to Faraday's law: $\mathcal{E} = -\frac{d\Phi_B}{dt}$. The induced current flows in whatever direction creates a magnetic field that opposes this flux change, following Lenz's law. Since the external field $\vec{B}$ is increasing through the loop, the induced current must flow in a direction that creates its own magnetic field opposing $\vec{B}$. This opposition doesn't prevent the flux change but minimizes it, consistent with nature's tendency to resist changes. Choice A is wrong because constant speed doesn't mean no flux change—the area of the loop inside the field is still increasing. Choice C contradicts Lenz's law; reinforcing the external field would accelerate rather than oppose the change. Choice D misunderstands the process—current flows throughout the entire time flux is changing, not just when motion stops. When studying electromagnetic induction, always ask: "What's changing?" Here it's the flux. Then apply Lenz's law: "The induced effect opposes the change." This systematic approach will help you tackle any induction problem by focusing on flux changes rather than getting distracted by constant speeds or forces.

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