Question 812: Electrostatics & Capacitance

Question

Consider a system of three charges $${q \over 3},{q \over 3}$$ and $$ - {{2q} \over 3}$$ placed at points A, B and C, respectively, as shown in the figure. Take O to be the centre of the circle of radius R and angle CAB = 60$$^\circ$$

Accepted Answer

Electric field at point O is vector sum of electric field due to all three point charges individually.
$$\overrightarrow E  = \overrightarrow {{E_A}}  + \overrightarrow {{E_B}}  + \overrightarrow {{E_C}} $$ and $$\overrightarrow {{E_A}}  =  - \overrightarrow {{E_B}} $$
So, $$\overrightarrow E  = \overrightarrow {{E_C}}  = {{2q/3} \over {4\pi {\varepsilon _0}.{R^2}}}$$ along $$\overrightarrow {OC} $$
$$ = {q \over {6\pi \varepsilon _0^2.{R^2}}}$$ along $$\overrightarrow {OC} $$
$$ = {{ + q'} \over {6\pi \varepsilon _0^2.{R^2}}}\widehat i$$
$$ = {{ - q'} \over {6\pi \varepsilon _0.{R^2}}}\widehat j$$
From geometry, we can find that $$\angle$$ABC = 30$$^\circ$$ and $$\angle$$ACB = 90$$^\circ$$
So, AB = 2R, AC = R, BC = $$\sqrt3$$R
Total potential energy of the system is,
$$U = {1 \over {4\pi {\varepsilon _0}}}\left[ {{{q/3 	imes q/3} \over {2R}} - {{q/3 	imes 2q/3} \over R} - {{q/3 	imes 2q/3} \over {\sqrt 3 R}}} ight] 
e 0$$
$$F = {{q/3 	imes q/3} \over {4\pi {\varepsilon _0} 	imes {{(\sqrt 3 R)}^2}}} = {{{q^2}} \over {54\pi {\varepsilon _0}{R^2}}}$$
Potential at point O is,
$$V = {{q/3 	imes q/3 - 2q/3} \over {4\pi {\varepsilon _0}R}} = 0$$

Answer

The electric field at point O is $${q \over {8\pi {\varepsilon _0}{R^2}}}$$ directed along the negative x-axis

Answer

The potential energy of the system is zero

Answer

The magnitude of the force between the charges at C and B is $${{{q^2}} \over {54\pi {\varepsilon _0}{R^2}}}$$

Answer

The potential at point O is $${q \over {12\pi {\varepsilon _0}R}}$$

Electrostatics & Capacitance

Consider a system of three charges ${q \over 3},{q \over 3} and - {{2q} \over 3} placed at points A, B and C, respectively, as shown in the figure. Take O to be the centre of the circle of radius R and angle CAB = 60^\circ

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