Electromagnetics, Waves, and Optics

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Physics › Electromagnetics, Waves, and Optics

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1

Figure 1

Three equal charges are at three of the corners of a square of side d. A fourth charge of equal magnitude is at the center of the square as shown in Figure above. Which of the arrows shown represents the net force acting on the charge at the center of the square?

B

A

C

D

Explanation

Because of the principles of superposition, each electric force that acts from the charges at the corners on to the charge at the center can be broken into components. Since all the charges are positive, all the forces will be repulsive. The forces acting from the top left and bottom right corners will cancel, leaving only the repulsive force coming from the bottom left corner.

2

Which of the following expressions gives the capacitance for a capacitor in a circuit in which the only factors known are a) the current through the circuit, b) the resistance of the circuit, and c) the charge that has accumulated on the capacitor?

Explanation

In this question, we're given a number of parameters of a circuit and are asked to find how we can use these various parameters to show the capacitance of the circuit's capacitor.

First, recall what a capacitor is; something that stores charge for a given voltage difference. In other words, when there is a voltage difference between the two plates of a capacitor, a certain amount of charge can be stored on these plates. The more charge that can be stored for a given voltage difference, the greater that capacitor's capacitance. This can be shown by the following equation.

From the above expression, is the capacitance which we are trying to find a proper expression for. is the charge accumulated on the plates, which is one parameter we're given. Voltage, , on the other hand, is not provided.

To put the voltage into different terms, we'll need to use Ohm's law, which states the following.

In other words, the voltage is proportional to both the current and the resistance of the circuit.

Since we are given both current and resistance as known parameters in the question stem, we can use these for our final answer. By substituting the in the capacitance expression with , we obtain the following answer.

3

What is the RMS voltage of a peak-to-peak AC current?

Explanation

This question requires two steps: the first is to calculate the peak voltage of the AC current, and the second requires conversion of the peak voltage to an RMS voltage. We are told that the current is peak-to-peak. Thus, the peak voltage is half of this, or . By definition, to convert peak voltage to RMS voltage, we must divide the peak voltage by :

4

You are given three resistors with known values:

You are asked to create a circuit with a total resistance of between and . How should you arrange the resistors to accomplish this?

and in parallel, connected to in series

, , and in parallel

and in parallel, connected to in series

and in parallel; is not necessary

, , and in series

Explanation

This question requires no math to correctly answer! You should not need to 'brute force' it. Although it is designed to appear time consuming, it should be relatively easily once the principle of resistors in parallel is understood. Whenever two resistors are connected in parallel, the net resistance must be less than the resistance of either of the two alone. When resistors are connected in series, the net resistance must be more than the resistance of either alone.

Explanation of correct answer:

and in parallel, connected to in series - It is possible to 'eyeball' this to see that this is at least feasible. and in parallel must make a network with an overall resistance less than . When added in series with (), the overall may fall between and . To confirm, one could do the math to calculate the overall resistance, but the point of this question is to use general principles to quickly eliminate the other, incorrect answer choices.

Explanations of incorrect answers:

, , and in parallel - This combination cannot possibly work since the overall resistance must be less than (the smallest resistor in parallel).

and in parallel, connected to in series - Regardless of the overall resistance of and in parallel, the connection with in series makes the total resistance more than .

and in parallel; is not necessary - Placing and in parallel must result in a resistance less than .

, , and in series - Connecting resistors in series results in an overall resistance greater than that of any one alone. Since and are included in series, the sum of the resistances is obviously much greater than what we are asked to produce and this choice can be immediately eliminated.

5

What size resistor should be connected across the terminals of a battery to produce a current of ?

Explanation

Ohm's law is .

In this case, and .

Solving for the resistance, , we get:

Substitute known values and solve for the unknown resistance:

6

For most people, the nearest distance that objects can be located away from the eye and still seen clearly is . This is referred to as the near point; if the object comes any closer, the object cannot be seen clearly. Suppose that a person who needs glasses cannot see objects clearly if they are closer than from the eye; that is to say, their near point is . A lens with what refractive power is needed in order to correct this person's vision to bring their near point to ?

Explanation

For this question, we're given the definition of near point. We're told what the near point is in the average person, and also the near point for a certain person who can't see well and needs glasses. We're asked to find the refractive power of a lens that will bring this individual's near point to the average, healthy value.

As was stated in the question stem, the near point is the closest distance of an object from the eye where that object can still be seen clearly. In the question, we're told that the normal value for this is . Moreover, a person with a near point of means that the object will need to be twice as far away, and no closer, to be seen clearly. Thus, in order to correct for this, a lens will be needed.

The idea is to be able to make the individual see things clearly when objects are located away. To accomplish this, a lens will need to diffract the light coming from an object away. This diffracted light will then need to form an image away, which is where this particular individual's near point is.

With this information in hand, we can use the lens-makers equation to solve for the refractive power of the lens.

We know that the object will be located a distance of away. Moreover, the image will need to form at a distance of away. However, since the image is forming on the same side of the lens that the object is located, the image will be virtual. Thus, the value used in the equation will be .

Also, remember that to find refractive power, we'll need to have our units be in meters.

Furthermore, recall that refractive power is equal to the inverse of focal length.

Hence, the power of the refractive lens will need to be diopters to correct this person's near point.

7

A light ray is traveling through air hits a transparent material at an angle of from the normal. It is then refracted at . What is the speed of light in the material?

Explanation

This problem requires Snell's Law and the corresponding equation:

We know that the index of refraction of air is:

We also know that:

and

Now we can plug in these values into the Snell's Law equation to find the index of refraction for the transparent material.

Finally, we need to calculate the speed of light moving through this transparent material now that we know the index of refraction for it. To do that, we need to use this equation:

Where is the speed of light and is the index of refraction. We plug in our known values and get:

8

What size resistor should be connected across the terminals of a battery to produce a current of ?

Explanation

Ohm's law is .

In this case, and .

Solving for the resistance, , we get:

Substitute known values and solve for the unknown resistance:

9

Suppose that a magnetic field is oriented such that it is pointing directly to the left, as in the picture shown below. If a positively charged particle were to begin traveling through this magnetic field to the right, in which direction would the particle's trajectory begin to curve?

Magnetic field

The particle would continue to move to the right unaffected

The particle would move out of the page

The particle would move down the page

The particle would move into the page

Explanation

In order to answer this question, it's important to understand the factors that determine the magnetic force experienced by a charge. We can begin by writing out the equation for magnetic force.

As shown in the above equation, the magnetic force is directly proportional to the particle's charge, its velocity, and the strength of the magnetic field itself. But, for the purposes of the this question, the most important factor is the angle of the particle's velocity with respect to the magnetic field.

Notice that if theta is equal to zero, then the sine of theta will be equal to zero as well. This, in turn, will cause the magnetic force to also be zero. This is also true if we were to define theta as .

Since the particle is moving in a direction that is parallel to the magnetic field lines but in the opposite direction, we have a situation in which theta is equal to . This means that the magnetic force on the particle is zero. As a result, the particle will continue to move through the magnetic field without changing its direction.

10

Suppose that car A and car B are both traveling in the same direction. Car A is going and is sounding a horn with a frequency of . If car B is traveling at a speed of in front of car A, what frequency of sound does car B hear?

Note: The speed of sound in air is .

Explanation

The important concept being tested in this question is the Doppler effect. When a source of sound waves (or any other wave) is emitted from a source and is moving relative to some observer, the actual frequency of the wave will be different for the observer.

For starters, we'll need to use the Doppler equation.

The trickiest part about this question is to decide whether to add or subtract in both the numerator and denominator. To find the right sign orientation, it's helpful to do a quick thought experiment. Imagine that only one of them is moving and the other is stationary. Decide how that will affect the observed frequency; will it increase it or decrease it? Then repeat for the other one.

First, let's consider the detector (aka observer). We're told that car B, the detector, is traveling ahead of car A and also in the same direction. This means that car B is driving away from car A. So in this situation, is the frequency that car B hears expected to increase or decrease? The answer is that it will decrease. Since car B is traveling away from car A, each successive wave will take longer to reach car B. Hence, we will use subtraction in the numerator because that will make the observed frequency smaller.

Now let's apply this same logic to the denominator, which deals with the source of the sound waves. We know that car A is traveling in the same direction as car B and is behind. This means that car A is traveling toward car B. So from this perspective, each successive wave is expected to get closer together, thus making the time between each wave smaller and the frequency bigger. In the denominator, will adding or subtracting make the observed frequency bigger? The answer is subtraction. By making a smaller number in the denominator, the entire fraction becomes larger.

Keeping this information in mind, we'll need to use subtraction in both the numerator and denominator. Once we plug in the values given in the question stem, we have everything we need to solve for the answer.

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