Paragraph: In a thin rectangular metallic strip a constant current $I$ flows along the positive…
Paragraph:
In a thin rectangular metallic strip a constant current $I$ flows along the positive $x$-direction, as shown in the figure. The length, width and thickness of the strip are $l, w$ and $d$, respectively.
A uniform magnetic field $\vec{B}$ is applied on the strip along the positive $y$-direction. Due to this, the charge carriers experience a net deflection along the $z$-direction. This results in accumulation of charge carriers on the surface $P Q R S$ and appearance of equal and opposite charges on the face opposite to $P Q R S$. A potential difference along the $z$-direction is thus developed. Charge accumulation continues until the magnetic force is balanced by the electric force. The current is assumed to be uniformly distributed on the cross section of the strip and carried by electrons.
Question:
Consider two different metallic strips ($1$ and $2$) of same dimensions (length $l$, width $w$ and thickness $d$ ) with carrier densities $n_{1}$ and $n_{2}$, respectively. Strip $1$ is placed in magnetic field $B_{1}$ and strip $2$ is placed in magnetic field $B_{2}$, both along positive $y$-directions. Then $V_{1}$ and $V_{2}$ are the potential differences developed between $K$ and $M$ in strips $1$ and $2$, respectively. Assuming that the current $I$ is the same for both the strips, the correct option(s) is(are)
If and , then
If and , then
If and , then
If and , then
Solution
$V = \frac{B I}{n e d}$
$\therefore V \propto \frac{B}{n}$
Or $\frac{V_1}{V_2} = \frac{B_1 n_2}{B_2 n_1}$
$\therefore$ if $B_1 = B_2$ and $n_1 = 2 n_2 \Rightarrow V_2 = 2 V_1$
And if $B_1 = 2 B_2$ and $n_1 = n_2 \Rightarrow V_2 = 0.5 V_1$
