Question

In FIGURE P31.33, a circular loop of radius $r$travels with speed $v$along a charged wire having linear charge density $I$. The wire is at rest in the laboratory frame, and it passes through the center of the loop.

a. What are $\overrightarrow{E}$and $\overrightarrow{B}$at a point on the loop as measured by a scientist in the laboratory? Include both strength and direction.

b. What are the fields $\overrightarrow{E}$and $\overrightarrow{B}$at a point on the loop as measured by a scientist in the frame of the loop?

c. Show that an experimenter in the loop’s frame sees a current$I=\lambda \nu$passing through the center of the loop.

d. What electric and magnetic fields would an experimenter in the loop’s frame calculate at distance r from the current of part c?

e. Show that your fields of parts b and d are the same.

Short answer

(a) Magnetic field $\overrightarrow{B}=\overrightarrow{0}$and Electric field $\overrightarrow{E}\left(r\right)=\frac{\lambda }{2\pi r_{\circ }}\hat{r}$

(b)$\overrightarrow{E_l}=\frac{\lambda }{2\pi r_{\circ }}\hat{r}$and $\overrightarrow{B_l}=-\frac{\lambda \nu }{2\pi r_{\circ }{c}^2}\hat{\theta }$

(c)$I=\lambda \nu$

(d)$\overrightarrow{B}\left(r\right)=-\frac{{\mu }_{\circ }I}{2\pi r}\hat{\theta }$

(e) Field at part b and d are same and that is$-\frac{{\mu }_{\circ }i}{2\mathrm{\pi r}}\hat{\theta }$

Step-by-step solution

Part (a) Step 1: Given information

We have been given that a circular loop of radius $r$travels with speed $v$along a charged wire having linear charge density $I$. The wire is at rest in the laboratory frame, and it passes through the center of the loop.

We need to calculate $\overrightarrow{E}$and $\overrightarrow{B}$at a point on the loop as measured by a scientist including both strength and direction.

Part (a) Step 2:  Simplify

In the lab frame, the charges will be stationary and that's why$\overrightarrow{B}=\overrightarrow{0}$

Now, using gauss law on a cylindrical surface around the line of charges with length $L$and radius $r$, the electric field at the loop is:$\overrightarrow{E}\left(r\right)=\frac{\lambda }{2\pi r_{\circ }}\hat{r}$

Part (b) Step 1: Given information

We have been given that a circular loop of radius $r$travels with speed $v$along a charged wire having linear charge density $I$. The wire is at rest in the laboratory frame, and it passes through the center of the loop.

We need to find the fields $\overrightarrow{B}$and $\overrightarrow{E}$on the loop measured by a scientist in the frame of the loop .

Part (b) Step 2:  Simplify

On transforming the field to the loop rest frame ,we get$$\begin{gathered} \overrightarrow{E_l}=\overrightarrow{E}+\overrightarrow{v}\times \overrightarrow{B}=\frac{\lambda }{2\pi r_{\circ }}\hat{r}+\overrightarrow{0}\frac{\lambda }{2\pi r_{\circ }} \\ \overrightarrow{B_l}=\overrightarrow{B}-\frac{1}{c^2}\hat{v}\times \overrightarrow{E} \end{gathered}$$

$=\overrightarrow{0}-\frac{v}{c^2}\hat{z}\times \frac{\lambda }{2\pi r_{\circ }}$

Where l- represent fields in the loop rest frame.

Part (c) Step 1: Given information

We have been given that a circular loop of radius $r$travels with speed $v$along a charged wire having linear charge density $I$. The wire is at rest in the laboratory frame, and it passes through the center of the loop.

We need to prove that an experimenter in loop's frame will sees a current $I=\lambda \nu$passing through the center of loop .

Part (c) Step 2: Prove

The current passing through the center of the loop is:$I=\frac{dq}{dt}=\frac{dq}{dx}\frac{dx}{dt}=\nu \lambda$

Hence ,the current is flowing in the $-\hat{z}$direction .

Part (d) Step 1: Given information

We have been given that a circular loop of radius $r$travels with speed $v$along a charged wire having linear charge density$I$. The wire is at rest in the laboratory frame, and it passes through the center of the loop.

We need to find the electric and magnetic fields calculated by an experimenter in the loop's frame at a distance $r$from the current of part c.

Part (d) Step 2: Simplify

On applying Ampere's law, we get

$\overrightarrow{B}\left(r\right)=-\frac{{\mu }_{\circ }I}{2\pi r}\hat{\theta }$

and,$\overrightarrow{E}=\overrightarrow{0}$

Part (e) Step 1: Given information

We have been given that a circular loop of radius $r$travels with speed $v$along a charged wire having linear charge density$I$. The wire is at rest in the laboratory frame, and it passes through the center of the loop.

We need to prove that our fields of parts b and d are the same.

Part (e) Step 2: Prove

It is given that $I=\lambda \nu$and $c^2=\frac{1}{{\mu }_{\circ }}$

We have,

localid="1650180996859" $Fieldatpartb=-\frac{\lambda \nu }{2\pi r_{\circ }{c}^2}\hat{\theta }=-\frac{1}{2\pi r}·\frac{1}{{\displaystyle \frac{1}{{\mu }_{\circ }}}}\hat{\theta }=-\frac{{\mu }_{\circ }I}{2\pi r}\hat{\theta }=Fieldatpartd$

Hence proved .