Question

A uniform magnetic field exists in a region given by $\vec{B}=3 \hat{i}+4 \hat{j}+5 \hat{k} T$. A rod of length $5\, m$ is placed along the $y$-axis and it is moved along the $x$-axis with constant speed $1\, ms ^{-1}$. Then induced e.m.f. in the rod will be

• A. zero
• B. $25 \, V$
• C. $20 \, V$
• D. $15 \, V$

Answer 1

Answer: (B)

We know that
Magnetic field vector $\vec{B}=3 \hat{i}+4 \hat{j}+5 \hat{k}$
length of rod $\vec{L}=5 \hat{j}$
velocity vector $\vec{v}=1 \hat{i}$
Induced emf is given as
$E=\vec{L} \cdot(\vec{v} \times \vec{B})$
$E=(5 \hat{j}) \cdot[(1 \hat{i}) \times(3 \hat{i}+4 \hat{j}+5 \hat{k})] $
$E=(5 \hat{j}) \cdot(-5 \hat{j}+4 \hat{k}) $
$E=-25 \,V$