AP Physics B & C - Multiple Choice Practice Questions on Electromagnetic Induction

“Maturity is often more absurd than youth and very frequently is more unjust to youth.”

– Thomas A. Edison

Michael Faraday’s discovery of electromagnetic induction was a turning point in the history of mankind. When he made the first public announcement that the relative motion between a magnet and a coil of wire could cause the flow of a feeble electric current through the coil, he had to face this question: “But what is the use?” Faraday countered this with another question: “What is the use of a new born baby?”

The baby has grown rapidly to become a very healthy youth who will remain so for many more decades!

The phenomenon responsible for the generation of electric power for feeding the modern world still continues to be electromagnetic induction.

Questions on electromagnetic induction are generally interesting. Click on the label ‘electromagnetic induction’ below this post; you will find all posts on electromagnetic induction published so far on this site.

Today we will discuss a few more multiple choice practice questions in this section.

(1) Earth’s resultant magnetic field at California has magnitude B tesla and it makes an angle θ with the horizontal. Assuming that there are no other magnetic fields, what will be the voltage induced between the tips of the wings of an airplane of wing-span L flying horizontally with speed v?

(a) BLv

(b) BLv sin θ

(c) BLv cos θ

(d) BLv/sin θ

(e) BLv/cos θ

Since the airplane is flying horizontally it can ‘cut’ the vertical magnetic field lines to generate a motional emf V given by

V = B_verticalLv where B_vertical is the vertical component of earth’s magnetic field at the place.

With reference to the adjoining figure we have

B_vertical = B sin θ

Therefore, the voltage induced between the tips of the wings of the airplane is BLv sin θ.

(2) A plane square loop of thin copper wire has 100 turns. Each side of the loop is 10 cm long and the loop is oriented with its plane making an angle of 30º with a uniform magnetic field of flux density 0.4 tesla. If the loop is rotated in 0.5 second so as to orient its plane at right angles to the magnetic field, what will be the magnitude of the average emf induced in the loop?

(a) 0.1 volt

(b) 0.2 volt

(c) 0.4 volt

(d) 0.8 volt

(e) 2 volt

The induced emf V is given by

V = – dФ/dt where dФ is the change in the total magnetic flux linked with the coil and dt is the time taken for the flux change.

[The negative sign is the consequence of Lenz’s law by which the induced emf has to oppose the change of flux dФ].

Since we are required just to find the magnitude of the induced voltage, we may ignore the negative sign

Since the coil has N (=100) turns, the total flux linked with the coil is Nφ where φ is the flux per turn given by

φ = BAcos θ where B = 0.4 tesla and A = area of the square loop = (0.1)² m² = 0.01 m²

The angle θ is the angle between the magnetic field and the area vector.

[Remember that the area vector is directed perpendicular to the plane of the coil].

Since the plane of the coil makes an angle of 30º with a magnetic field, the area vector makes an angle of 60º with the magnetic field.

The initial magnetic flux linkage is NBAcos 60º = 100×0.4×0.01×(1/2) = 0.2 weber.

Since the area vector and the magnetic field are finally parallel (or anti-parallel), the final flux linkage is NBAcos 0º = 100×0.4×0.01 = 0.4

The change of flux dФ = 0.4 – 0.2 = 0.2

Therefore, induced emf = (Change of flux) /(Time) = 0.2/0.5 = 0.4 volt.

(3) Suppose that the resistance (R) of the loop in the above question is 10 Ω. What will be the induced current in the loop if the loop is kept stationary and the magnetic field is steadily reduced to zero in a time of 40 millisecond?

(a) 0.2 A

(b) 0.5 A

(c) 1 A

(d) 1.5 A

(e) 2 A

The initial magnetic flux linked with the loop (as shown above) is NBAcos 60º = 100×0.4×0.01×(1/2) = 0.2 weber.

When the magnetic field is reduced to zero, the magnetic flux is reduced to zero. Therefore the change of magnetic flux is 0.2 weber. The emf V induced in the loop is given by

V = (Change of flux) /(Time) = 0.2/(40×10^–3) volt = 5 volt.

The current induced in the loop is V/R = 5/10 A = 0.5 A [Option (b)]

The following question is meant specifically for AP Physics C aspirants:

(4) A straight conductor of length L and mass M can slide down along a pair of long, smooth, conducting vertical rails P and Q of negligible resistance (Fig.). A resistor of resistance R is connected between the ends of the rails as shown in the figure. A uniform magnetic field of flux density B acts perpendicularly into to the plane containing the rails and the sliding conductor. The terminal velocity of fall of the rod is

(a) MgR/LB

(b) mgL/B²R²

(c) B²L²/mgR

(d) mgB/L²R

(e) mgR/B²L²

When the rod slides down under gravity, the magnetic flux linked with the closed circuit comprising the rod, rails and the resistor R changes and a current is induced in the circuit. The induced emf is the motional emf BLv where v is the velocity of the rod. The induced current I in the circuit is BLv/R.

By Lenz’s law the induced current has to oppose the motion of the rod. It is the magnetic force ILB which brings in this opposition. When the velocity of the rod increases, the opposing magnetic force also increases. When the magnitudes of the gravitational force (weight Mg of the rod) and the opposing magnetic force become equal, the rod moves with a constant (terminal) velocity.

Therefore, we have

ILB = Mg

Substituting for I we have (BLv/R)LB = Mg

Or, B²L²v/R = Mg

This gives v = MgR/B²L²

Now, let me ask you a question:

If the direction of the magnetic field in the above question is reversed, will the rod still attain a terminal velocity? Think of it and arrive at the answer ‘YES’.

You will find a few more questions (with solution) in this section here.

AP Physics B & C - Multiple Choice Practice Questions Involving Electromagnetic Induction

“You must not lose faith in humanity. Humanity is like an ocean; if a few drops of the ocean are dirty, the ocean does not become dirty.”

– Mahatma Gandhi

Questions involving electromagnetic induction have been discussed on several occasions on this site. You can access all of them either by trying a search for ‘electromagnetic induction’ using the search box provided on this page or by clicking on the label ‘electromagnetic induction’ below this post. Questions on electromagnetic induction are generally interesting and so the aspirants of AP Physics B as well as AP Physics C will definitely ‘enjoy’ them. The following questions are meant for gauging the depth of your knowledge and understanding in this area:

(1) A particle of mass m and charge –q moves with uniform speed from point A to point B along a straight path AOB in the positive y-direction. A plane conducting circular loop is arranged near the path AOB (as shown in the figure) so that the loop and the path AOB are in the same plane. Pick out the correct statement regarding the current induced in the loop during the interval the charged particle moves from A to B:

(a) The induced current increases and flows clockwise in the loop

(b) The induced current increases and flows anticlockwise in the loop

(c) The induced current increases, reaches a maximum and then decreases; but it flows clockwise in the loop

(d) The induced current increases, reaches a maximum and then decreases; but it flows anticlockwise in the loop

(e) The induced current changes its direction

This question may confuse many among you quite a lot. The particle has a negative charge and that too may enhance your doubts.

Think of the moving charge as a moving current element. Since the charge is negative and is moving along the positive y-direction, the direction of the conventional current is along the negative y-direction. At the circular loop this current element gives rise to a magnetic field perpendicular to the plane of the loop, directed towards the reader. When the charged particle is at A, the magnetic flux through the loop is small and when it is nearest to the circular loop (at O), the magnetic flux through the loop is maximum. The increasing magnetic flux induces a current in the loop. By Lenz’s law the induced current must flow in the clockwise sense (so that the magnetic flux produced by the induced current opposes the increasing magnetic flux produced by the moving charge).

When the charge moves from O to B, the magnetic flux linked with the loop decreases. The current induced in the loop now changes its direction since the magnetic flux produced by the induced current has to oppose the decreasing magnetic flux produced by the moving charge. The correct option is (e).

[The last option can be further refined and stated as follows:

(e) The induced current flows in the clockwise sense and anticlockwise sense respectively when the charged particle traces the paths AO and OB].

(2) A rectangular conducting loop PQRS (Fig.) is located in a magnetic field B which acts perpendicularly into the plane of the loop. The magnetic field is constant and it exists over a wide region, well beyond the extent of the loop. The loop is moved, within the magnetic field, with uniform velocity v in a direction mutually perpendicular to the direction of the magnetic field and the side QR. Pick out the correct statement regarding the induced current (if any):

(a) The induced current in the side PQ flows from P to Q.

(b) The induced current in the side RS flows from S to R.

(c) There is no induced current in the loop since the voltage induced in the side PS as well as QR is zero.

(d) There is no induced current in the loop since no voltage is induced in any of the sides of the loop.

(e) There is no induced current in the loop since the voltages induced in the sides PS and QR are equal and they act in opposition in the loop.

No current is induced in the loop since the magnetic flux linked with the loop is the same in all positions of the loop. So there is no change of flux to produce an induced emf in the loop. Even though the last three options indicate that the induced current in the loop is zero, the reason for this is correct in the case of option (e) only. Equal motional emfs are generated in the sides PS and QR and these act in the loop as two voltage sources in opposition.

(3) A copper rod moves at a constant velocity in a direction perpendicular to its length. A constant magnetic field mutually perpendicular to the rod and its direction of motion exists in the region. Pick out the correct statement from the following:

(a) The electric potential at the centre of the rod is maximum when the rod moves.

(b) The electric potential at the centre of the rod is minimum when the rod moves.

(c) The electric potential is the same at all points of the rod when it moves.

(d) An electric field is produced in the rod when it moves.

(e) The electric potential is zero at the centre of the rod and increases towards the ends

When the rod moves at right angles to the magnetic field, the free electrons in the rod experience magnetic (Lorentz) force and shift towards one end. Thus there is an accumulation of negative charges at one end of the rod and consequently there exists an electric field in the rod [Option (d)].

The following question is specifically for AP Physics C aspirants:

(4) The series combination of an inductance L and a resistance R shown in the adjoining figure is part of an electric circuit which drives a current i through the combination. When the current through the combination is 6 mA and is decreasing at the rate of 10 mA s^–1, the potential difference between points A and B is 6 mV. When the current through the combination is 6 mA and is increasing at the rate of 10 mA s^–1, the potential difference between points A and B is 12 mV. The values of the inductance L and resistance R are respectively

(a) 4 H and 6 Ω.

(b) 3 H and 6 Ω

(c) 0.5 H and 3.5 Ω

(d) 0.3 H and 1.5 Ω

(e) 1.5 H and 1.5 Ω

It is the potential difference between the points A and B that drives a current through L and R. When the current in the circuit decreases, the inductor has to try to oppose the decrease by inducing a voltage in it to aid the 6 mV across the points A and B. Therefore Kirchhoff’s voltage equation for this case is

– L(di/dt) + iR = P.D. between A and B

[We could have better written this as (P.D. between A and B) + L(di/dt) = iR]

Substituting known values, we have

– L×10×10^–3 + 6×10^–3 R = 6×10^–3

Or, – L×10 + 6R = 6 -----------(i)

When the current in the circuit increases, the inductor has to try to oppose the increase by inducing a voltage in it to oppose the 12 mV across the points A and B. Therefore Kirchhoff’s voltage equation for this case is

L(di/dt) + iR = P.D. between A and B

[We could have better written this as (P.D. between A and B) – L(di/dt) = iR]

Substituting known values, we have

L×10 + 6R = 12 -----------(ii)

Adding Eq(i) and Eq(ii) we have 12 R = 18 from which R = 1.5 Ω.

On substituting the value of R in Eq(ii) we obtain L = 0.3 H [Option (d)].

AP Physics C – Answer to Free Response Practice Question on Electromagnetic Induction

A free response practice question on electromagnetic induction was posted on 10^th October 2010. As promised, I give below a model answer (along with the question) for your benefit.

[You can access earlier posts in this section either by clicking on the label 'electromagnetic induction' below this post or by trying a search for 'electromagnetic induction' using the search box provided on this page].

Two infinitely long straight parallel wires W₁ and W₂, separated by a distance ‘a’ in free space, carry equal currents I flowing in opposite directions as shown in the adjoining figure. A square loop PQRS of side ‘a’, made of nichrome wire of resistance ρ Ω per metre is arranged with its plane lying in the plane of the wires W₁ and W₂ so that the sides PQ and RS of the loop are parallel to the wires W₁ and W₂. The side PQ of the loop is at a distance ‘a’ from the wire W₂. Now, answer the following questions in terms of the given quantities and fundamental constants:

(a) Determine the magnetic flux density at a point midway between the wires W₁ and W₂.

(b) Determine the magnetic flux density at the mid point of the square loop PQRS.

(c) Calculate the magnetic flux through the loop PQRS.

(d) What is the average emf induced in the loop when the current through the wires is switched off in a time of 50 ms?

(e) When the current through the wires is switched off, it is found that at a certain instant t, the current decays at the rate of 40 As^–1. Calculate the current induced in the loop PQRS at the instant t.

Indicate the direction of the current in the loop and justify your answer.

The magnitude of the magnetic flux density B at a point distant a from an infinitely long straight conductor carrying current I is given by

B = μ₀I/2πa where μ₀ is the magnetic permeability of free space

The magnetic flux density at a point midway between the wires W₁ and W₂ is the resultant of the magnetic flux densities produced by these wires. Each wire produces magnetic flux density of magnitude μ₀I/2πa.

These fields are directed perpendicular to the plane containing the wires and outwards (towards the reader) and hence they add up to produce a resultant flux density of magnitude μ₀I/2πa + μ₀I/2πa= μ₀I/πa.

[Since the magnetic flux density is a vector, you should not forget to mention its direction].

(b) At the mid point of the square loop PQRS the magnetic fields due to the wires W₁ and W₂ are directed perpendicular to the plane containing the wires. But the field due to the wire W₂ is directed into the plane of the figure (away from the reader) where as the field due to the wire W₁ is directed outwards (towards the reader). The resultant field is directed into the plane of the figure (away from the reader) since the wire W₂ produces stronger field.

The magnitude of the resultant magnetic flux density at the mid point of the square loop PQRS is [(μ₀I)/(2π×3a/2)] – [(μ₀I)/(2π×5a/2)]

This is equal to (μ₀I/πa)[(1/3) – (1/5)] = 2μ₀I/15πa

(c) To find the magnetic flux through the loop PQRS, consider a strip of very small width dx at distance x from the wire W₂ as shown in the figure. The resultant magnetic flux density B at the strip is given by

B = (μ₀I/ 2π) [1/x – 1/(a+x)]

Magnetic flux linked with the strip = B dA = Badx

[dA is the area of the strip of length a and width dx]

Magnetic flux Ф linked with the entire loop PQRS is given by

Ф = _a∫^2a Badx = (μ₀Ia/2π) _a∫^2a [1/x – 1/(a+x)]dx

[The limits of integration are x = a and x = 2a].

Therefore, Ф = (μ₀Ia/2π) [ln x – ln (a+x)] between limits x = a and x = 2a.

Or, Ф = = (μ₀Ia/2π) ln [x/(a+x)] between limits x = a and x = 2a.

= (μ₀Ia/2π) [ln(2/3) – ln(1/2)]

= (μ₀Ia /2π) ln (4/3) ...............................(i)

[The unit of magnetic flux density is tesla (or, weber per mrtre²) and the unit of magnetic flux is weber].

(d) The average emf V_average induced in the loop is given by

V_average = Rate of change of magnetic flux

= Change of magnetic flux/ Time

= [(μ₀Ia /2π) ln(4/3) – 0]/(50×10^–3) since the current through the wires is switched off in a time of 50 ms

Therefore, V_average = = (10μ₀Ia /π) ln (4/3)

(e) The emf V induced in the loop PQRS at the instant t is given by

V = – dФ/dt, the negative sign appearing because of Lenz’s law.

Ignoring the negative sign, we have from equation (i)

V = dФ/dt = [(μ₀a /2π) ln (4/3)](dI/dt)

Here dI/dt = 40 As^–1, as given in the question.

Substituting for dI/dt, we have V = (20μ₀a /π) ln (4/3)

The induced current in the loop = V/R where R is the resistance of the loop, which is equal to 4aρ ohm.

Therefore, induced current = (20μ₀a /4πaρ) ln (4/3) = (5μ₀ /πρ) ln (4/3) ampere.

The direction of the induced current in the loop is clockwise, as indicated in the figure.

The resultant magnetic field is directed normally into the plane of the loop and is decreasing when the current is switched off. The induced current should oppose this change and should therefore produce a magnetic field acting in the same direction (normally into the plane of the loop). This is made possible by the clockwise flow of the induced current.

AP Physics C - Free Response Practice Question on Electromagnetic Induction

“Non-violence leads to the highest ethics, which is the goal of all evolution. Until we stop harming all other living beings, we are still savages”

– Thomas A. Edison

Today I will give you a free response practice question on electromagnetic induction. You may try to answer this question within 15 minutes. Here is the question:

Two infinitely long straight parallel wires W₁ and W₂, separated by a distance ‘a’ in free space, carry equal currents I flowing in opposite directions as shown in the adjoining figure. A square loop PQRS of side ‘a’, made of nichrome wire of resistance ρ Ω per metre is arranged with its plane lying in the plane of the wires W₁ and W₂ so that the sides PQ and RS of the loop are parallel to the wires W₁ and W₂. The side PQ of the loop is at a distance ‘a’ from the wire W₂. Now, answer the following questions in terms of the given quantities and fundamental constants:

(a) Determine the magnetic flux density at a point midway between the wires W₁ and W₂.

(b) Determine the magnetic flux density at the mid point of the square loop PQRS.

(c) Calculate the magnetic flux through the loop PQRS.

(d) What is the average emf induced in the loop when the current through the wires is switched off in a time of 50 ms?

(e) When the current through the wires is switched off, it is found that at a certain instant t, the current decays at the rate of 40 As^–1. Calculate the current induced in the loop PQRS at the instant t.

Indicate the direction of the current in the loop and justify your answer.

This question carries 15 points. Try to answer it. I’ll be back soon with a model answer for your benefit.