Electricity & Magnetism: Induced E.M.F. & Current

[zorro]

Homework Statement

2 parallel long, straight conductors lie on a smooth plane surface. 2 other parallel conductors rest on them at right angles so as to form a square of side 'a' initially. A uniform magnetic field B exists at right angles to the plane containing the conductors. Now they start moving out with constant velocity v

a)Will the induced e.m.f. be time dependent?
b)Will the current be time dependent?

The Attempt at a Solution

The change in flux through the loop at any time 't' is given by
dΦ/dt = B x a x dx/dt where x is the length of the conductor inside the magnetic field.
so E = -Badx/dt = -Bav

As velocity is constant (time independent), the e.m.f. is time independent.

The current induced = -Bav/R which is also time independent.

The answer does not exactly match this

 

[kuruman]

Two questions.

1. When you say "they start moving", who is "they"? Two conductors or all four?
2. What is R? Specifically, what happens to R when "they start moving"?

 

[zorro]

1. I think all 4 start moving. I don't have a reason for it though.
2. R is the resistance of the conductors.

The answer says that emf induced is time dependent where as current is independent of time

 

[kuruman]

Does the resistance R in your expression change as the conductors move? If so, how?

 

[zorro]

why would it change? I don't have any reason for it to change.
It depends on the length of the conductor which is constant.

 

[kuruman]

why would it change? I don't have any reason for it to change.
It depends on the length of the conductor which is constant.

And how long is the conducting path along which the current runs?

 

[zorro]

OK I understood
So the resistance is given by
R = r(a + 2x), where r is the resistance per unit length.
What next?

 

[kuruman]

Is the resistance constant?

 

[zorro]

No it is not. You can see it depends on the length of the conductor inside the magnetic field.

 

[kuruman]

So if the induced emf is time-independent and the resistance depends on time, what does that make the current?

 

[zorro]

If I see the problem your way, current is time-dependent.

However, the answer given is -
Induced EMF- time dependent
Induced Current- time independent

 

[kuruman]

If I see the problem your way, current is time-dependent.

However, the answer given is -
Induced EMF- time dependent
Induced Current- time independent

The given answer makes no sense. Sometimes given answers are incorrect and I think this is one such case.

 

[zorro]

I referred the solution manual and this is what is given:

a) Yes
instantaneous flux Φ = B(a + 2vt)²
therefore E = dΦ/dt = 4Bv(a+2vt)

b) No,
instantaneous current, i = E/R

Now R = 4(a+2vt)r where r = resistance per unit length
therefore i = 4Bv(a+2vt)/4r(a+2vt) = Bv/r which is a constant.

Hence the current will be time independent.

I could not understand this. Please explain if you understood.

 

[kuruman]

The answer in the manual is consistent with both pairs of bars moving relative to each other. Since you were not sure, I initially assumed that only one pair is moving. You can figure this out on your own. This is what you do.

1. Find an expression for the side of the square as a function of time t once the bars start moving.
2. Find an expression for the area of the square as a function of time t.
3. Use Faraday's Law to find the induced emf. This should answer question (a).
4. Find an expression for the perimeter of the square along which current runs as a function of time.
5. Find the resistance of the square R as a function of time.
6. Find the induced current using I = emf/R. This should answer question (b).

 

[zorro]

I never did such a problem when all four conductors move relative to each other.
Its clear now anyways.
Thanks!