Solving for x in Circular Current Equation

[Jaccobtw]

Homework Statement

A very large loop of metal wire with radius 1m is driven with a linearly increasing current at a rate of 200A/s. A very small metal wire loop with radius 5cm is positioned a small distance away with its center on the same axis (the loops are coaxial). The small loop experiences an induced emf of 983nV. What is the separation of the loops in m? Note that a subtraction step in the solution makes it sensitive to significant figures. Keep at least four figures in your calculation.

Relevant Equations

$$\int E \cdot ds = \varepsilon = -\frac{d}{dt}\int B \cdot dA$$
$$B = \frac{\mu_0 I R^{2}}{2(x^{2} + R^{2})^{3/2}}$$

The second equation gives the magnetic field at a point away from the center of a circular current. If we multiply this by the the area we get a function for the magnetic flux. We have an increasing current which induces an increasing magnetic field. Now just solve for x in the second equation to get the distance the second loop is away from the first loop. But I don't think you can just plug in a changing current in for I. Where have I gone wrong? Thank you

 

[TSny]

Relevant Equations:: $$\int E \cdot ds = \varepsilon = -\frac{d}{dt}\int B \cdot dA$$ $$B = \frac{\mu_0 I R^{2}}{2(x^{2} + R^{2})^{3/2}}$$

The second equation gives the magnetic field at a point away from the center of a circular current. If we multiply this by the the area we get a function for the magnetic flux. We have an increasing current which induces an increasing magnetic field. Now just solve for x in the second equation to get the distance the second loop is away from the first loop. But I don't think you can just plug in a changing current in for I.

You can't solve for the value of $x$ using the second equation since you don't know the value $B$ or $I$.
However, you are given the value of the induced emf. Can you use the first equation to solve for the value of $x$?

 

[Delta2]

You should work towards calculating the expression for the EMF on the small loop. It will turn out that it is a function of x (and $\frac{dI}{dt}$ and that's where the rate of change of current comes in). You wrote that you multiply B by the area to get the flux, that's correct, but what you should do to the flux expression to get the expression for the EMF?

 

[Jaccobtw]

You should work towards calculating the expression for the EMF on the small loop. It will turn out that it is a function of x (and $\frac{dI}{dt}$ and that's where the rate of change of current comes in). You wrote that you multiply B by the area to get the flux, that's correct, but what you should do to the flux expression to get the expression for the EMF?

if you integrate the changing current with respect to time, you'll find that I = 200t. the problem is now that I have two variables for the magnetic field - t and x. How do I take the integral of B with respect to both those variables IF that's what I should do?

 

[Delta2]

if you integrate the changing current with respect to time, you'll find that I = 200t. the problem is now that I have two variables for the magnetic field - t and x. How do I take the integral of B with respect to both those variables IF that's what I should do?

You want to keep x as unknown cause that's what the problem is asking for. The time t will be simplified when you take the time derivative of the flux. We assume that the loop is small enough so that the B-field is homogeneous inside it. No need to do spatial integration to find the flux, a simple multiplication of the B-field at the center with the area of the small loop should suffice.

 

[Jaccobtw]

You want to keep x as unknown cause that's what the problem is asking for. The time t will be simplified when you take the time derivative of the flux. We assume that the loop is small enough so that the B-field is homogeneous inside it. No need to do spatial integration to find the flux, a simple multiplication of the B-field at the center with the area of the small loop should suffice.

The problem is when I try to isolate x on one side I keep getting a negative inside a square root.

 

[Delta2]

The problem is when I try to isolate x on one side I keep getting a negative inside a square root.

Hmmm, do you get the correct answer if you assume that the induced EMF is -983nV instead of the 983nV that the problem states?

 

[Jaccobtw]

Hmmm, do you get the correct answer if you assume that the induced EMF is -983nV instead of the 983nV that the problem states?

$$\varepsilon = -\frac{d}{dt} BA$$ 1
$$\frac{\varepsilon}{A} = -\frac{dB}{dt}$$ 2
$$\int\frac{\varepsilon dt}{A} = -\int dB = -B$$ 3
$$\frac{\varepsilon t}{A} = -\frac{\mu_0 200t R^{2}}{2(R^{2} + x^{2})^{3/2}}$$ 4
$$\varepsilon = -\frac{\mu_0 200 R^{2} A}{2(R^{2} + x^{2})^{3/2}}$$ 5, t cancels
$$(R^{2} + x^{2})^{3/2} = -\frac{\mu_0 200 R^{2} A}{2\varepsilon }$$ 6
After this I take both sides to the 2/3 power. Then because R has a value of 1, I subtract 1 from both sides then take the sqaure root of both sides. Because 1 is bigger than what it is subtracting, the value ends up being negative on the right side inside a square root. Not sure what I'm doing wrong

 

[kuruman]

I think your approach is fundamentally flawed. When you write $\varepsilon = -\frac{d}{dt} BA$, you take $A$ to be the area of the big loop. That is inappropriate. It is the changing magnetic flux at the position of the smaller loop that generates the emf. So in the above equation $B$ is the magnetic field at the location of the smaller loop and $A$ must be the area of the smaller loop. As @Delta2 already noted, you may assume that the magnetic field is uniform over the area of the loop and has its on-axis value.

Also note that the negative sign in front of the rate of change of flux is there because of Lenz's law. It indicates that the induced emf always opposes the rate of change of magnetic flux. If you apply Faraday's law formally with vectors and use the right hand rule to link the directed normal to the loop with the circulation of the line integral, you should have no problem with negative signs under radicals. If you are lazy like me, you can just take the absolute value and ignore the negative sign.

 

[Jaccobtw]

My mistake. I wrote the permeability of free space as $4 \times 10^{-7}$ instead of $4\pi \times 10^ {-7}$. All of that leads to the correct answer. Thanks a lot