Question

The figure shows a metallic solid block, placed in a way so that its faces are parallel to the coordinate axes. Edge lengths along axis $x,y$ and $z$ are $a,b$ and $c$ respectively. The block is in a region of uniform magnetic field of magnitude $30mT$. One of the edge length of the block is $25cm$. The block is moved at $4m/s$ parallel to each axis and in turn, the resulting potential difference $V$ that appears across the block is measured.
(1)When the block is moved at $4m/s$ parallel to the $y$- axis, $V=24mV$,
(2)when the block is moved at $4m/s$ parallel to the $z$- axis $V=36mV$,
(3)when the block is moved at $4m/s$ parallel to the $x$- axis, $V=0$
Using the given information, correctly match the dimensions of the block in column-I with the values given in column-II.
$\begin{array}{cc}\text{column I}& \text{Column II}\\ \text{(A) a}& \text{(p) 20 cm}\\ \text{(B) b}& \text{(q) 24 cm}\\ \text{(C) c}& \text{(r) 25 cm}\\ \text{(D)}\frac{bc}{a}& \text{(s) 30 cm}\\ & \text{(t) 26 cm}\end{array}$

• A. $A\to p,B\to q,C\to s,D\to t$
• B. $A\to q,B\to s,A\to r,D\to p$
• C. $A\to r,B\to s,C\to p,D\to q$
• D. $A\to s,B\to p,C\to s,D\to t$

Answer 1

The correct option is C $A\to r,B\to s,C\to p,D\to q$

In general emf generated,
$\epsilon =\overrightarrow{B}.(\overrightarrow{l}\times \overrightarrow{v});$
$(\overrightarrow{l}\times \overrightarrow{v})$ is component of $\overrightarrow{l}$ perpendicular to $\overrightarrow{v}$.

Since when $\overrightarrow{v}$ is along x - axis, $\epsilon =0,\overrightarrow{B}$ must be along x - axis.

When $\overrightarrow{v}$ is along y - axis,
emf generated $\epsilon =B\left(cv\right)=24mV\text{(Since}c\text{swipes the defining flux)}\Rightarrow 30\times {10}^{-3}\times c\times 4=24\times {10}^{-3}\Rightarrow c=0.2\Rightarrow c=20cm$

When $\overrightarrow{v}$ is along z- axis
emf generated $\epsilon =B\left(bv\right)=36mV\text{(Since}b\text{swipes the defining flux)}\Rightarrow 30\times {10}^{-3}\times b\times 4=36\times {10}^{-3}\Rightarrow b=0.3m\Rightarrow b=30cm$
$\Rightarrow b=30cm,c=20cmanda=25cm\Rightarrow \frac{bc}{a}=\frac{30\times 20}{25}=24cm$