Faraday's Law: Induced Voltage and Current in a Loop with Resistance R

[horsewnoname]

Homework Statement

An infinitely long wire carries current $I=I_0sin(wt)$. A distance $a$ from this wire is an $w$ by $l$ loop with resistance $R$ with induced voltage $V$ and induced current $i$. Find the induced voltage and current in the loop.

Homework Equations

Faraday's law is given by $\varepsilon = \oint \mathbf{E}\cdot d\mathbf{l}=-\frac{d\phi }{dt}$where $\phi$ is the magnetic flux given by $\int \boldsymbol{B}\cdot d\boldsymbol{s}$.

The Attempt at a Solution

I know that in order to find the voltage, I need to find the emf which is given by Faraday's law stated above. I pursue this by first coming up with an expression for the magnetic flux.

Although, to do this, I first need to know the magnetic field produced by the infinite wire. From memory (or by Ampere's law), I know this to be $B=\frac{\mu _0I}{2\pi r}$.

The problem I am having is that the magnetic field is a function of $r$ which bothers me. When I use this magnetic field expression to determine the flux and then take the derivative with respect to time to yield the emf, the result is an expression for the voltage that varies with $r$ which makes no sense. What am I forgetting?

 

[NFuller]

Homework Statement

An infinitely long wire carries current $I=I_0sin(wt)$. A distance $a$ from this wire is an $w$ by $l$ loop with resistance $R$ with induced voltage $V$ and induced current $i$. Find the induced voltage and current in the loop.

I assume the $w$ in $\text{sin}(wt)$ is supposed to be different than the $w$ in the loop dimensions. I also assume that the loop is in the plane of the wire, is it or is it perpendicular to the wire?

The problem I am having is that the magnetic field is a function of rrr which bothers me. When I use this magnetic field expression to determine the flux and then take the derivative with respect to time to yield the emf, the result is an expression for the voltage that varies with rrr which makes no sense. What am I forgetting?

You have to integrate $\mathbf{B}$ across the area of the loop in order to find $\phi$.

 

[horsewnoname]

I assume the $w$ in $\text{sin}(wt)$ is supposed to be different than the $w$ in the loop dimensions. I also assume that the loop is in the plane of the wire, is it or is it perpendicular to the wire?

You have to integrate $\mathbf{B}$ across the area of the loop in order to find $\phi$.

Yes, of course - I just need to integrate the magnetic field over the dimensions of the loop so that I can get an expression for flux in terms of the given quantities. I feel foolish for not realizing this! And yes, you're assumptions were correct. The $w$ in the sine should actually be an omega - that was my typo. Should I edit my question for posterity or will your stated assumptions suffice?

 

[NFuller]

Should I edit my question for posterity or will your stated assumptions suffice?

It should be fine since you have clarified it.