Question

dx Tr Figure B4 gs wire carrying a steady current I. The axis of the rod is maintained perpendicular to the wire with the near end a distance r away from it. dx is an element of the rod at a distance x. (a) Use Ampére's law to find the magnetic field at position x and determine its direction (6 marks) (b) Find the magnitude of the emf induced in dx. (2 marks) (c) Find the magnitude of the emf induced in the entire rod and determine its polarity. (7 marks)

Answer 1

a)

Let B be magnetic field due to current I
At any r+x distance,
B * 2$\pi$ * (r+x) = $\mu_o$ I

B = $\mu_o$ I / 2$\pi$ *( r+ x)
Direction is inside the paper.

b)

Emf induced in dx length with velocity v = B* dx* v
= $\mu_o$I / [2$\pi$ *(r+x)] * dx* v

c)
$\int_{r}^{r+l}\frac{\mu_o * I }{ 2 \pi *(r+x) } v dx$ = $\frac{\mu_o * I*v }{ 2 \pi } \int_{r}^{r+l}\frac{1}{(r+x)}dx$ = $\frac{\mu_o * I*v }{ 2 \pi } ln(r+x) |_{r}^{r+l}$

=$\frac{\mu_o * I*v }{ 2 \pi } ln \frac{2r+l}{2r}$

Higher potential on left or positive polarity on left and negative polarity on right.