Magnetism and Motion of Charged Particles (4C) - Chem & Physics

1. Identify the relationship between $\vec{F}$ and $\vec{v}$ for magnetic forces on a moving charge.

   $\vec{F}\perp\vec{v}$ (magnetic force does no work). The magnetic force is always perpendicular to velocity due to the cross product, ensuring zero work done as $\vec{F}\cdot\vec{v}=0$.

2. State the formula for the magnetic force magnitude on a straight wire of length $L$ carrying current $I$.

   $F = ILB\sin\theta$. The force on a current-carrying wire integrates the Lorentz force over charge carriers, depending on current, length, field, and angle.

3. State the magnitude of the magnetic field at distance $r$ from a long straight wire carrying current $I$.

   $B = \frac{\mu_0 I}{2\pi r}$. The field strength decreases inversely with distance, derived from Ampere's circuital law for an infinite straight wire.

4. Identify the condition on the velocity direction that makes the magnetic force on a charge equal to zero.

   $\theta = 0^\circ$ or $180^\circ$ (\,$\vec{v}\parallel\vec{B}$\,). Magnetic force vanishes when velocity is parallel or antiparallel to the magnetic field because $\sin\theta=0$ at these angles.

5. State the cyclotron period for a nonrelativistic charge in a uniform magnetic field.

   $T = \frac{2\pi m}{|q|B}$. The period is the time for one circular orbit, derived from angular frequency as $T=2\pi/\omega$.

6. State the SI unit of magnetic field $B$ in base units.

   $1\,\text{T} = \frac{\text{N}}{\text{A}\cdot\text{m}}$. The tesla is defined from the force on a current-carrying wire, expressed in base SI units as newtons per ampere-meter.

7. How does the magnetic force direction change for a negative charge compared with a positive charge?

   It is opposite to $\vec{v}\times\vec{B}$. A negative charge reverses the force direction due to the sign of $q$ in the Lorentz force equation.

8. What is the vector form of the magnetic force on a moving charge (Lorentz magnetic force)?

   $\vec{F} = q\,\vec{v}\times\vec{B}$. The vector form of the Lorentz magnetic force arises from the cross product of velocity and magnetic field vectors, scaled by the charge.

9. What is the pitch of the helical path for a charge with velocity component $v_{\parallel}$ along $\vec{B}$?

   $\text{pitch} = v_{\parallel}T = \frac{2\pi m v_{\parallel}}{|q|B}$. The pitch represents axial advance per cyclotron period, combining parallel velocity with the orbital period in the magnetic field.

10. Identify the direction of motion for a positive charge entering uniform $\vec{B}$ into the page with $\vec{v}$ to the right.

    Upward (force toward top of page). For positive charge with velocity rightward and field into the page, the Lorentz force directs upward via the right-hand rule, causing initial upward deflection.

11. State Faraday's law for the induced emf in a loop in terms of changing magnetic flux.

    $\mathcal{E} = -\frac{d\Phi_B}{dt}$. Faraday's law states that induced emf equals the negative rate of change of magnetic flux through the loop.

12. State the formula for the magnetic force magnitude on a charge moving in a magnetic field.

    $F = |q|vB\sin\theta$. The magnitude of the magnetic force on a moving charge is derived from the Lorentz force law, incorporating the charge, velocity, magnetic field strength, and the sine of the angle between velocity and field vectors.

13. State the magnetic field magnitude inside an ideal long solenoid with turn density $n$ and current $I$.

    $B = \mu_0 n I$. Inside a solenoid, the uniform field results from the additive contributions of tightly wound current loops, proportional to turn density and current.

14. State the cyclotron angular frequency for a nonrelativistic charge in a uniform magnetic field.

    $\omega = \frac{|q|B}{m}$. Cyclotron frequency emerges from equating magnetic force to centripetal force, independent of velocity for nonrelativistic cases.

15. State the expression for the radius of circular motion of a charge moving perpendicular to uniform $B$.

    $r = \frac{mv}{|q|B}$. The radius balances centripetal force with magnetic force for perpendicular motion, yielding $r = \frac{mv}{|q|B}$.

16. Identify the speed selector condition for undeflected motion through crossed fields $\vec{E}\perp\vec{B}$.

    $v = \frac{E}{B}$. In crossed fields, undeflected motion occurs when electric and magnetic forces balance, so $qE = qvB$.

17. Identify the direction of motion for an electron entering uniform $\vec{B}$ out of the page with $\vec{v}$ upward.

    Deflects to the left. An electron's negative charge reverses the force direction, leading to leftward deflection when velocity is upward and field is out of the page.

18. What is the magnetic flux through a flat surface of area $A$ in a uniform field $B$?

    $\Phi_B = BA\cos\theta$. Magnetic flux quantifies the component of the magnetic field passing through the surface, given by the dot product of field and area vectors.

19. What is the direction of the magnetic force on a positive charge relative to $\vec{v}$ and $\vec{B}$?

    Along $\vec{v}\times\vec{B}$ (right-hand rule). The direction follows from the cross product in the Lorentz force, determined by the right-hand rule for positive charges.

20. Find the required speed for no deflection if $E=300\,\text{N/C}$ and $B=0.10\,\text{T}$ in a velocity selector.

    $v = 3.0\times 10^3\,\text{m/s}$. Speed for no deflection satisfies $v=E/B$, resulting in $3.0\times 10^3$ m/s from given field strengths.

21. What is the direction of the magnetic field around a long straight wire with current (right-hand rule)?

    Circles around wire; thumb $I$, fingers give $\vec{B}$. The right-hand rule for a straight wire follows from Ampere's law, with magnetic field lines encircling the current.

22. State Lenz's law in one sentence about the direction of induced current.

    Induced current opposes the change in $\Phi_B$. Lenz's law, a consequence of energy conservation, dictates that induced current creates a field opposing the flux change.

23. What is the vector form of the magnetic force on a current-carrying wire segment?

    $\vec{F} = I\,\vec{L}\times\vec{B}$. The vector force on a wire segment results from the cross product of the length vector in the direction of current and the magnetic field.

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