Question 585: Electrostatics & Capacitance

Question

A three coulomb charge moves from the point $(0,-2,-5)$ to the point $(5,1,2)$ in an electric field expressed as $\vec{E}=2 x \hat{\mathrm{i}}+3 \mathrm{y}^2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}} \mathrm{N} / \mathrm{C}$. The work done in moving the charge is $\_\_\_\_$ J.

Accepted Answer

The work done (W) by an external agent in moving a charge (q) in an electric field ( $\overrightarrow{\mathrm{E}}$ ) is given by the change in potential energy, which is the negative of the work done by the field.
The electric force acting on the charge by electric field is $\overrightarrow{\mathrm{F}}=\mathrm{q} \overrightarrow{\mathrm{E}}$
So, the work done is :
$$ \mathrm{W}=\mathrm{q} \int \overrightarrow{\mathrm{E}} \cdot \mathrm{~d} \overrightarrow{\mathrm{r}} $$
Where $\mathrm{d} \overrightarrow{\mathrm{r}}=\mathrm{dxÃÂ®}+\mathrm{dy} \hat{\mathrm{j}}+\mathrm{dz} \hat{\mathrm{k}}$ is the infinitesimal displacement vector.
$$ \mathrm{W}=\mathrm{q} \int\limits_{(0,-2,-5)}^{(5,1,2)}\left(2 \mathrm{x} \hat{\imath}+3 \mathrm{y}^2 \hat{\jmath}+4 \hat{\mathrm{k}}ight) \cdot(\mathrm{dx} \hat{\imath}+\mathrm{dy} \hat{\jmath}+\mathrm{dz} \hat{\mathrm{k}}) $$
$$\Rightarrow $$ $\mathrm{W}=\mathrm{q}\left[\int_0^5 2 \mathrm{xdx}+\int_{-2}^1 3 \mathrm{y}^2 \mathrm{dy}+\int_{-5}^2 4 \mathrm{dz}ight]$
$$\Rightarrow $$ $\mathrm{W}=\mathrm{q}\left[2\left[\frac{\mathrm{x}^2}{2}ight]_0^5+3\left[\frac{\mathrm{y}^3}{3}ight]_{-2}^1+4[\mathrm{x}]_{-5}^2ight]$
$$\Rightarrow $$ $\mathrm{W}=3\left[\left[5^2-0^2ight]+\left[1^3-(-2)^3ight]+4[2-(-5)]ight]$
$$\Rightarrow $$ $\mathrm{W}=3 	imes(25+9+28) $
$$\Rightarrow $$ $ \mathrm{W}=3 	imes 62=186 \mathrm{~J}$
Therefore, the work done in moving the charge is 186 J.

Electrostatics & Capacitance

A three coulomb charge moves from the point (0,-2,-5) to the point (5,1,2) in an electric field expressed as \vec{E}=2 x \hat{\mathrm{i}}+3 \mathrm{y}^2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}} \mathrm{N} / \mathrm{C}. The work done in moving the charge is \_\_\_\_ J.

Integer / Numeric Answer Type