Question 297: Electrostats & Capacitance

Question

A point charge of $10^{-8} \mathrm{C}$ is placed at origin. The work done in moving a point charge $2 \mu \mathrm{C}$ from point $A(4,4,2) \mathrm{m}$ to $B(2,2,1) \mathrm{m}$ is $\_\_\_\_$ J. $\left(\frac{1}{4 \pi \epsilon_{\mathrm{o}}}=9 	imes 10^9ight.$ in SI units)

Accepted Answer

The work done ( W ) by an external agent in moving a charge q from point A to point B in the field of a source charge $Q$ is equal to the change in electrostatic potential energy :
$$ \mathrm{W}=\mathrm{U}_{\mathrm{B}}-\mathrm{U}_{\mathrm{A}}=\mathrm{q}\left(\mathrm{~V}_{\mathrm{B}} -\mathrm{V}_{\mathrm{A}} ight) $$
Where the electrostatic potential V at a distance r from a source charge Q is :
$$ V=\frac{1}{4 \pi \epsilon_0} \frac{Q}{r} $$
The source charge at origin is, $\mathrm{Q}=10^{-8} \mathrm{C}$
The test charge is $\mathrm{q}=2 \mu \mathrm{C}=2 	imes 10^{-6} \mathrm{C}$
The distance $r$ of a point $(x, y, z)$ from the origin is $r=\sqrt{x^2+y^2+z^2}$
For Point A (4, 4, 2):
$$ r_A=\sqrt{4^2+4^2+2^2}=\sqrt{16+16+4}=\sqrt{36}=6 \mathrm{~m} $$
For Point B (2, 2, 1):
$$ r_B=\sqrt{2^2+2^2+1^2}=\sqrt{4+4+1}=\sqrt{9}=3 \mathrm{~m} $$
So, the potential at $\mathrm{A}\left(\mathrm{V}_{\mathrm{A}}ight)$ is,
$$ V_A=\frac{k Q}{r_A}=\frac{\left(9 	imes 10^9ight) 10^{-8}}{6}=\frac{90}{6}=15 V $$
And the potential at $\mathrm{B}\left(\mathrm{V}_{\mathrm{B}}ight)$ is,
$$ V_B=\frac{k Q}{r_B}=\frac{\left(9 	imes 10^9ight) 10^{-8}}{3}=\frac{90}{3}=30 V $$
So, the work done by external agent is,
$$ \mathrm{W}=\mathrm{q}\left(\mathrm{~V}_{\mathrm{B}}-\mathrm{V}_{\mathrm{A}}ight) $$
$$\Rightarrow $$ $\mathrm{W}=\left(2 	imes 10^{-6}ight)(30-15)$
$$\Rightarrow $$ $$ \mathrm{W}=30 	imes 10^{-6} \mathrm{~J} $$
Therefore, the work done to be $30 	imes 10^{-6} \mathrm{~J}$.
Hence, the correct option is (A).

Answer

$30 	imes 10^{-6}$

Answer

0

Answer

$15 	imes 10^{-6}$

Answer

$45 	imes 10^{-6}$

Electrostats & Capacitance

A point charge of 10^{-8} \mathrm{C} is placed at origin. The work done in moving a point charge 2 \mu \mathrm{C} from point A(4,4,2) \mathrm{m} to B(2,2,1) \mathrm{m} is \_\_\_\_ J. \left(\frac{1}{4 \pi \epsilon_{\mathrm{o}}}=9 \times 10^9\right. in SI units)

Choose the correct answer: