Time Dependent Current in a wire

[kjlchem]

Homework Statement

An infinite straight wire carries a current I that varies with time as shown above. It increases from 0 at t = 0 to a maximum value I1 = 2.1 A at t = t1 = 14 s, remains constant at this value until t = t2 when it decreases linearly to a value I4 = -2.1 A at t = t4 = 24 s, passing through zero at t = t3 = 21.5 s. A conducting loop with sides W = 20 cm and L = 57 cm is fixed in the x-y plane at a distance d = 49 cm from the wire as shown.

What is ε1, the induced emf in the loop at time t = 7 s? Define the emf to be positive if the induced current in the loop is clockwise and negative if the current is counter-clockwise.

Homework Equations

B = μI/2∏r

Flux = B*A

-dflux/dt = ε

The Attempt at a Solution

I don't understand what I'm doing wrong with this problem.

This is what I have so far...

(dB*A)/dt= ε, A = L(W)

μ(dI)(L)W/(2∏rdt) = ε

μ=12.566*10^-7
dI = 2.1 A
L = .57 m
W = .2 m
dt=14 s.

On the left side of the box, r = .49 m and the current is negative, so the emf is positive.
On the right side of the box, r = 1.06 m and the current is positive, so the emf is negative.
Putting the 2 emf's together by subtracting the right side from the left side, I get an emf of -3.753*10^-9V.

What am I doing wrong?

 

[Redbelly98]

On the left side of the box, r = .49 m and the current is negative, so the emf is positive.
On the right side of the box, r = 1.06 m and the current is positive, so the emf is negative.
Putting the 2 emf's together by subtracting the right side from the left side, I get an emf of -3.753*10^-9V.

What am I doing wrong?

I'm not sure about the method you are using. For calculating EMF for a straight section of wire, I am only familiar with doing that for the wire moving through a magnetic field.

Since the wire loop is not moving, I think you have to use ε=-dflux/dt instead. So first I would first calculate the flux through the loop -- as a function of time, during the time interval that contains 7 s.

 

[kjlchem]

Yeah, I used ε = -dflux/dt. The flux as a function of time = μ(dI)(L)W/(2∏rdt)

 

[Redbelly98]

Yeah, I used ε = -dflux/dt. The flux as a function of time = μ(dI)(L)W/(2∏rdt)

That won't work here; for one thing, there is no r given in this problem. I think I see your problem though:

Flux = B*A

That only works if B is uniform over the whole area. It isn't; B is stronger at the side of the rectangle closest to the wire, and weaker at the far side.

Instead, you'll need to do an integral to calculate the flux:

Flux = $\int B \cdot dA$​

 

[asteriatic]

Your approach to the problem is correct, but there are a few errors in your calculations. Here is a step-by-step solution to help you understand where you went wrong:

1. First, we need to find the magnetic field at the location of the loop, which is at a distance of 49 cm from the wire. Using the equation B = μI/2∏r, we get:

B = (4∏*10^-7)(2.1)/((2∏)(0.49)) = 1.357*10^-5 T

2. Next, we need to find the flux through the loop, which is given by the equation Flux = B*A. The area of the loop is L*W = (0.57)(0.2) = 0.114 m^2. Therefore, the flux through the loop is:

Flux = (1.357*10^-5)(0.114) = 1.548*10^-6 Wb

3. Now, we can use the equation -dflux/dt = ε to find the induced emf at time t = 7 s. We know that the flux is changing linearly between t = 0 and t = 14 s, so we can use the formula for a linear change in flux:

ΔFlux/Δt = (1.548*10^-6)/14 = 1.106*10^-7 Wb/s

4. Finally, we can plug this value into the equation -dflux/dt = ε to find the induced emf at t = 7 s:

-ε = (1.106*10^-7)/(-7) = 1.58*10^-8 V

Since the current in the wire is increasing at t = 7 s, the induced current in the loop will be clockwise, making the induced emf positive. Therefore, the value of ε1 at t = 7 s is 1.58*10^-8 V.

Hope this helps clarify the solution for you!