Current flowing in a cylinder and induced EMF

[Jahnavi]

Homework Statement

Homework Equations

The Attempt at a Solution

The current will decrease , as a result an EMF will be induced in the cylinder .

The EMF induced E = -dΦ/dt

I am assuming magnetic field through the cylinder to be same as that at the center of the a current carrying coil which is given by μ₀i/2r .

E = -μ₀πrdi/(2dt)

I tried to calculate equivalent resistance .

I think we can consider thin cylinder material as a thin sheet of breadth 'l' , length '2πr' and thickness 'd' .

It can be divided into thin wires of length '2πr' , width dx and thickness 'd'

Current is flowing across the breadth .

Consider a very thin section of wire of width dx at a distance x across the breadth from one end .

All these thin wires are in parallel .

Their equivalent resistance turns out to be ρ2πr/ld .

So , the resistance across which the current is flowing is R = ρ2πr/ld .

The current equation would be iR + E = 0 .

iR - μ₀πrdi/(2dt) = 0

ρ2πri/ld - μ₀πrdi/(2dt) = 0

Is the equation correct ?

 

[Chandra Prayaga]

Why is the current equation iR + E = 0? Should it not be iR = E?

 

[TSny]

I am assuming magnetic field through the cylinder to be same as that at the center of the a current carrying coil which is given by μ₀i/2r .

No. Think solenoid.

So , the resistance across which the current is flowing is R = ρ2πr/ld .

That looks good to me.

The current equation would be iR + E = 0 .

iR - μ₀πrdi/(2dt) = 0

As @Chandra Prayaga pointed out, there seems to be a sign problem here. Note that your equation would imply that di/dt is positive; i.e., the current would increase instead of decrease.

 

[Jahnavi]

Thank you TSny .

No. Think solenoid.

You are right

Magnetic field inside a solenoid is μ₀ni .

How would I find 'n' of the equivalent solenoid ? 'n' is the number of terms per unit length .

 

[Lord]

Solenoid consists of wires. This cylinder has a continuous current distribution thus evaluate using Ampere's circuital law.

B.2πr = μ0
That gives B =μ0i/2πr

 

[TSny]

Note that in $\mu_0ni$, $\, i$ is the current in one turn. Try to interpret the meaning of $ni$.

 

[Jahnavi]

Note that in $\mu_0ni$, $\, i$ is the current in one turn. Try to interpret the meaning of $ni$.

Wow !

I didn't think in this way .

Magnetic field inside cylinder would be μ₀i/l .

This gives us option 2) .

Is that correct ?

 

[TSny]

Magnetic field inside cylinder would be μ₀i/l .

This gives us option 2) .

Yes. Good work.

 

[Jahnavi]

Thank you so much