# Induced EMF in Non-Uniform Magnetic Field

• arl146
In summary: Since the wire is moving in a magnetic field, it will experience a force. The direction of this force is determined by the q charge inside the wire. q will move in the direction of the force and the side of the wire where it resides will get a positive (+) charge and the side where it resides will get a negative (-) charge.
arl146

## Homework Statement

A square wire loop of dimensions L X L (L = 15.5 cm) oriented in the xy-plane and centered on the x-axis enters a region where the magnetic field (B = 0.855 T) is first oriented in the +z- and then in the -z-direction as shown in the diagram below. (Note: W = 15.501 cm and the magnetic field is zero outside the shaded area.) The loop moves at speed v = 4.55 ms in the +x-direction. In the questions that follow, when asked for the emf, a positive emf will be defined as an emf induced counterclockwise (if viewed from above the loop); and vice versa.

(i.) What is the induced emf in the loop when the center of the loop is at x = -W?
(ii.) What is the induced emf in the loop when the center of the loop is at the origin?
(iii.) What is the induced emf in the loop when the center of the loop is at x = +W/2?
(iv.) What is the induced emf in the loop when the center of the loop is at x = +W?

See below

## The Attempt at a Solution

I think if I get help on one of them, then I should be able to figure out the others.
I'm not too sure how to approach this one, but I was thinking to use E=Blv or the flux=BA equation. But I'm not sure how that works out

no one can help me with this ? =/

still looking for help ...

In general emf = vBL for these sorts of situations but a drawing would be nice.

As WatermelonPig said, EMF=vBl
and its polatiry can be found out by F=q vXB (cross product)

bt we need a diagram to understand the question

Oh, sorry, wasnt paying attention that I didn't have one attached!

#### Attachments

• prob14a.gif
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(i) when center will be at x=-w ... which part of loop will be inside mag. field area and which part will induce emf (for this, you should have studied motional emf)

only half of the loop will be inside the magnetic field that is pointing up. the left side half isn't in a B field. So just the right side that's in the magnetic field induces emf?

yes, that's correct :)

so, what does that mean haha i still am not sure what to do for it

you did first
now for second, you need to find polarity of induced emf in on the 2 wires (left and right) of loop ...

for this use the formula of force on moving charge in mag. field, $\vec{F} \ = \ q \ \vec{v} \times \vec{B}$

As you would know, the wire moving in mag. field will act like a battery with a (+) and a (-) pole.
suppose there is a positive charge in wire moving with wire ... find direction force on it (generally along wire in these kind of questions). the side where + charge moves is (+) and other is (-)

thus find net emf in wire in case (ii)

so wait .. i don't get it, how did i do the first one?

and for (ii.) how do you use that without q ?

Last edited:
Since you're dealing with a loop, it would be more straightforward to calculate the rate of change of the magnetic flux through the loop to find the emf.

motional EMF in a wire moving in mag field is vBL , here v is velocity B is mag field and L is the length.

cupid.callin said:
motional EMF in a wire moving in mag field is vBL , here v is velocity B is mag field and L is the length.

i did that already it was wrong

is it sign problem or magnitude is wrong?

Oh wait, I got it now :)
Okay, for the second: i don't get what youre saying for that ..

Can I please have some help on the other one's then?

do you know how to find polarity of the charges in a wire?

i don't think ... I am not too sure what that means

arl146 said:
i don't think ... I am not too sure what that means

When a wire will move in mag. field, EMF will be induced in it.

so there will be a current flowing in wire and thus one end will get positive charge and other negative.

Can you find which end will get which charge?

.. i have no idea?

you know of formula for force on a charge in magnetic field ... F = q (v X B)

lets suppose there is a positive q charge inside the wire ... This charge will feel a mag. force when wire is moving ... find the direction of this force ... q will move in direction of Force and the side of wire where it moves will get (+) and other (-)

wouldnt the force just be in the direction opposite of the motion, so. then the left side would be positive and the right side would be negative

is that right?

Can someone help with this problem as well? =/ I am lost

I actually just need help on the second one. I got the rest!

Let x be the position of the front end of the loop. What's the flux through the loop in terms of B, L, and x?

I actually just got it ! It was twice my first answer but positive. I'm not too sure why it was twice, could you explain that?

You're the one who's supposed to be able to explain how you got the answer!

haha i know, but i don't really know how i got it. i just kinda guessed because all else failed. it doesn't make sense to me... half was in the + B-field and half was in the - B-field I would just think that they'd cancel out and it would equal 0.

The flux through the loop is indeed 0, but the emf doesn't depend on the flux but on the rate of change of the flux. That's what you need to calculate.

does it have anything to do with how that person was talking abour charges? like how the left side say wants to pull more to the left and the right side wants to pull more towards the right and so that puts twice the change on it ?

hello, what is the difference between the flux density of air gap and iron in magnetic circuit.
let say i have an iron ring with an air gap of 1mm. do the air gap and iron ring produce the same magnitude of flux density and flux.

## 1. What is induced EMF in a non-uniform magnetic field?

Induced EMF (electromotive force) in a non-uniform magnetic field refers to the creation of an electric current in a conductor due to a changing magnetic field. This change in magnetic field can be caused by the motion of the conductor through the magnetic field or by a change in the strength of the magnetic field itself.

## 2. How is induced EMF in a non-uniform magnetic field different from a uniform magnetic field?

In a uniform magnetic field, the strength and direction of the magnetic field is the same at all points. This means that the induced EMF will be constant throughout the conductor. In a non-uniform magnetic field, the strength and direction of the magnetic field varies, resulting in a non-constant induced EMF along the conductor.

## 3. What factors affect the magnitude of induced EMF in a non-uniform magnetic field?

The magnitude of induced EMF in a non-uniform magnetic field is affected by the rate of change of the magnetic field, the strength of the magnetic field, and the length and orientation of the conductor in the magnetic field. The larger the rate of change and the stronger the magnetic field, the greater the induced EMF will be.

## 4. How is Lenz's Law related to induced EMF in a non-uniform magnetic field?

Lenz's Law states that the direction of the induced EMF will be such that it opposes the change in magnetic flux that caused it. In a non-uniform magnetic field, this means that the induced EMF will create a current that produces a magnetic field in the opposite direction of the changing magnetic field, thus opposing the change.

## 5. What are some practical applications of induced EMF in non-uniform magnetic fields?

Induced EMF in non-uniform magnetic fields is used in a variety of technologies, such as generators, transformers, and electric motors. It is also the principle behind induction cooking, where a changing magnetic field is used to heat up a conductive cooking vessel.

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