Question

A metal rod of length L slides horizontally at constant speed v on frictionless insulating rails through a region of uniform upward magnetic field of magnitude B (Figure 20.124).

On a diagram, show the polarization of the rod and the direction of the Coulomb electric field inside the rod. Explain briefly. What is the magnitude of the Coulomb electric field inside the rod? What is the potential difference across the rod? What is the emf across the rod? What are the magnitude and direction of the force you have to apply to keep the rod moving at a constant speed v?

Short answer

(i) $E=BV$

(ii)The diagram is drawn below.

(iii)$e=BLV$

(iv)$F=\frac{B^2{L}^{2}V}{R}$

Step-by-step solution

Given Data

$\begin{array}{l}\text{Length=}L\\ \text{Speed}=v\\ \text{Magnetic\hspace{0.33em}Field}=B\end{array}$

Concept

The phenomenon occurred because of the wave nature of the radiation of the electromagnetic force is known as polarization.

Step 3(i): Find the magnitude of the Coulomb electric field inside the rod 

At equilibrium of electron inside the rod,

$\begin{array}{c}eE=eVB\\ E=BV\end{array}$

Hence, the magnitude of the Coulomb electric field inside the rod $E=BV$

Step 4(ii): Find the potential difference across the rod

Since, electron experiences force in upward direction means “electric field” is in downward direction .Therefore from A to B as shown in figure.

Step 5(iii): Find the emf across the rod

emf, (electric field= potential difference)

$\begin{array}{l}e=EL\\ e=BLV\end{array}$

Hence, the emf across the rod is$e=BLV$

Step 6(iv): Find the force

Force,

$\begin{array}{c}F=BIL\\ =B\left(\frac{e}{R}\right)L\\ =B\left(\frac{BLV}{R}\right)L\\ F=\frac{B^2{L}^{2}V}{R}\end{array}$

Hence the force is$F=\frac{B^2{L}^{2}V}{R}$