Question

An infinite sheet carrying a uniform surface charge density o lies on the $x y$-plane. The work done to carry a charge $q$ from the point $A =a(\hat{ i }+2 \hat{ j }+3 \hat{ k })$ to the point $B =a(\hat{ i }-2 \hat{ j }+6 \hat{ k })$ (where $a$ is a constant with the dimension of length and $\varepsilon_{0}$ is the permittivity of free space) is

• A. $\frac{3\sigma aq}{2\varepsilon_{0}}$
• B. $\frac{2\sigma aq}{\varepsilon_{0}}$
• C. $\frac{5\sigma aq}{2\varepsilon_{0}}$
• D. $\frac{3\sigma aq}{\varepsilon_{0}}$

Answer 1

Answer: (A)

$\frac{3\sigma aq}{2\varepsilon_{0}}$

Answer 2

The given
$A =a(\hat{ i }+2 \hat{ j }+ 3 \hat{ k })$
$B =a(\hat{ i }-2 \hat{ j }+6 \hat{ k })$
$A B = O B - O A$
$=a(\hat{ i }-2 \hat{ j }+6 \hat{ k })-a(\hat{ i }+2 \hat{ j }+ 3 \hat{ k })$
$A B =a(-4 \hat{ j }+3 \hat{ k })$
Work done $=q\left(\frac{\sigma}{2 \varepsilon_{0}}\right) \hat{ k } \cdot A B$ (along to $Z$-axis)
$=q\left(\frac{\sigma}{2 \varepsilon_{0}}\right) \hat{ k } \cdot a(-4 \hat{ j }+3 \hat{ k })=\frac{3 q \sigma a}{2 \varepsilon_{0}}$
$(\because \hat{ i } \cdot \hat{ i }=\hat{ j } \cdot \hat{ j }=\hat{ k } \cdot \hat{ k }=1$ and $\hat{ i } \cdot \hat{ j }=\hat{ j } \cdot \hat{ k }=\hat{ k } \cdot \hat{ i }=0)$