Question

A $1\mu A$ beam of protons with a cross-sectional area of $0.5\text{sq.mm}$ is moving with a velocity of $3\times {10}^{4}m{s}^{-1}$. Then charge density of beam is

• A. $6.6\times {10}^{-4}{\text{C/m}}^3$
• B. $6.6\times {10}^{-5}{\text{C/m}}^3$
• C. $6.6\times {10}^{-6}{\text{C/m}}^3$
• D. None of these

Answer 1

The correct option is B $6.6\times {10}^{-5}{\text{C/m}}^3$

Given:
$I={10}^{-6}A$
$A=5\times {10}^{-7}{\text{m}}^2$
$v=3\times {10}^4\text{m/s}$
Charge density = $\frac{\text{charge}}{\text{volume}}$
$\frac{It}{Al}=\frac{I}{Av}$
$=\frac{{10}^{-6}}{5\times {10}^{-7}\times 3\times {10}^4}$
$=\frac{100}{15}\times {10}^{-5}$
$=6.6\times {10}^{-5}{\text{C/m}}^3$