Question

Learning Goal:

To understand the terms in Faraday's law and to be able to identify the magnitude and direction of induced emf.

Faraday's law states that induced emf is directly proportional to the time rate of change of magnetic flux. Mathematically, it can be written as

E=???B?t,

where E is the emf induced in a closed loop, and

??B?t

is the rate of change of the magnetic flux through a surface bounded by the loop. For uniform magnetic fields the magnetic flux is given by ?B=B? ?A? =BAcos(?), where ? is the angle between the magnetic field B? and the normal to the surface of area A.

To find the direction of the induced emf, one can use Lenz's law:

The induced current's magnetic field opposes the change in the magnetic flux that induced the current.

For example, if the magnetic flux through a loop increases, the induced magnetic field is directed opposite to the "parent" magnetic field, thus countering the increase in flux. If the flux decreases, the induced current's magnetic field has the same direction as the parent magnetic field, thus countering the decrease in flux.

Recall that to relate the direction of the electric current and its magnetic field, you can use the right-hand rule: When the fingers on your right hand are curled in the direction of the current in a loop, your thumb gives the direction of the magnetic field generated by this current.

1)

Find the flux ?B through the loop.

Express your answer in terms of x, y, and B.

2)

If the magnetic field steadily decreases from B to zero during a time interval t, what is the magnitude E of the induced emf?

Express your answer in terms of x, y, B, and t

3.

If the magnetic field steadily decreases from B to zero during a time interval t, what is the magnitude I of the induced current?

Express your answer in terms of x, y, B, t, and the resistance R of the wire.

4.

If the magnetic field steadily decreases from B to zero during a time interval t, what is the direction of the induced current?

If the magnetic field steadily decreases from to zero during a time interval , what is the direction of the induced current?

clockwise

counterclockwise 5. Part F Which of the following changes would result in a clockwise emf in the loop? When you consider each option, assume that no other changes occur. Check all that apply. Check all that apply. The magnitude of B? increases. The magnitude of B? decreases. The loop rotates through 45 degrees about the vertical axis (vertical dotted line) shown in the diagram. The loop rotates through 45 degrees about the horizontal axis (horizontal dotted line) shown in the diagram. The loop moves to the right while remaining in the plane of the page. The loop moves toward you, out of the page, while remaining parallel to itself.

Answer 1

1)

Let B be the magnetic field passing through a rectangular loop of length x and breadth y

Assuming the plane of the rectangular loop is perpendicular to the direction of magnetic field, so perpendicular of the area of the loop will be parallel to the direction of magnetic field i.e   $\small \theta$ = 0 degree , so cos $\small \theta$ = 1

so, Magnetic flux, $\small \phi$ = B.A

= Bxy  

2)

Change in the magnetic flux, $\small \Delta \phi$ =xy( 0 - B )= - xyB

and time interval, dt = t-0 = t

EMF, $\small \varepsilon = - \Delta \phi$ /dt

= - (-xyB) / t = Bxy / t

3)

As calculated above, Emf induced, $\small \varepsilon$ = Bxy / t

Resistance of wire is R

so, By Ohm's Law :

V = IR

$\small \Rightarrow$ Bxy / t = IR

so, induced current, I = Bxy / Rt

4)

Since the magnetic flux is decreasing dut to decreasing magnrtic field from B to zero, then direction of Induced current in such a way that it oppose the cause of , so it will be in direction of magnetic field.so, that it can increase flux By using Right hand rule,

If magnetic filed is going into the page, then direction of induced current will be in clockwise.

If magnetic field is coming out of the page, then direction of induced current will be in anticlockwise,

5)

Assuming the direction of magnetic field is coming out of the page,

To introduce the EMF in clockwise, the change in magnetic field B should be positive , so that its cause could be opposed by induced EMF such as induced magnetic field will going into the page, and by using Right Hand rule, the direction of induced EMF will be clockwise.

so, Ony (a) increase the magnetic field is correct.