Name: Continuous Charge Distribution
Uploaded: 2019-07-19
Duration: 5 min 39 s
Description: This video will help us understand the necessity of continuous charge distribution and learn about the concept of linear charge density. Learn more.

Question

Three charges are placed on the vertices of an equilateral triangle of side length $I .$ Find the sign and magnitude of charge $q$ placed at centroid so that system of charges will be in equilibrium.

Accepted Answer

The charge at any vertex will remain in equilibrium if the net force experienced by this charge due to all other three charges is zero.
Let $Q$ be the required charge to be kept at the centroid G.

Considering the charge at A,
Force $F_{1}$ on charge at A due to charge at B.

${\vec{F}}_{1} = \frac{1}{4 π \in_{\circ}} \frac{q^{2}}{a^{2}} a l o n g B A$

Force $F_{2}$ on charge at A due to charge at C

${\vec{F}}_{2} = \frac{1}{4 π \in_{\circ}} \frac{q^{2}}{a^{2}} a l o n g C A$

Since angle between $F_{1}$ and $F_{2}$ is $60^{\circ}$ .

${\vec{F}}_{1} + {\vec{F}}_{2} = \sqrt{3} \frac{1}{4 π \in_{\circ}} \frac{q^{2}}{a^{2}} a l o n g G A$

Also, the distance of centroid G from any vertex is $\frac{a}{\sqrt{3}}$

The nature of charge to be kept at G has to be opposite (−ve) so that it exerts a force of attraction on charge ( $+ q$ ) kept at A to balance the force ${\vec{F}}_{1} + {\vec{F}}_{2}$

Force exerted by ( $- Q$ ) kept at G on charge ( $+ q$ ) at A $= \frac{1}{4 π \in_{\circ}} \frac{Q q}{{(\frac{a}{\sqrt{3}})}^{2}} = \frac{1}{4 π \in_{\circ}} . \frac{Q 3 q}{a^{2}} a l o n g A G$

Equating the two forces, being equal and opposite.

$\sqrt{3} \frac{1}{4 π \in_{\circ}} \frac{q^{2}}{a^{2}} = \frac{1}{4 π \in_{\circ}} \frac{3 Q q}{a^{2}} \Rightarrow Q = - \frac{q}{\sqrt{3}}$

Three charges are placed on the vertices of an equilateral triangle of side length $I .$ Find the sign and magnitude of charge $q$ placed at centroid so that system of charges will be in equilibrium.

Important Questions on Electric Charges and Fields

Learn and Prepare for any exam you want !

Three charges are placed on the vertices of an equilateral triangle of side length I. Find the sign and magnitude of charge q placed at centroid so that system of charges will be in equilibrium.

Important Questions on Electric Charges and Fields

Learn and Prepare for any exam you want !

Three charges are placed on the vertices of an equilateral triangle of side length $I .$ Find the sign and magnitude of charge $q$ placed at centroid so that system of charges will be in equilibrium.