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 point charges of $+ 2 μ C, - 3 μ C$ and $- 3 μ C$ are kept at the vertices $A, B$ and $C$ respectively of an equilateral triangle of side $20 cm$ as shown in figure. What should be the sign and magnitude of the charge to be placed at the mid-point $(M)$ of side $B C$ so that the charge at $A$ remains in equilibrium?

Accepted Answer

Net electric field at A due to charges at B and C is

Let charge placed at M be $q_{M}$ . The forces acting on charge ( $q_{A} = + 2 µ C$ ) are $F_{A B}$ , $F_{A C}$ and $F_{A M}$ as shown in figure.

${\vec{F}}_{A B} = \frac{1}{4 {πε}_{\circ}} \frac{q_{A} q_{B}}{{r_{A B}}^{2}} a l o n g \vec{A B}$

${\vec{F}}_{A B} = 9 \times 10^{9} \times \frac{(2 \times 10^{- 6}) (3 \times 10^{- 6})}{{(0.20)}^{2}} a l o n g \vec{A B}$

${\vec{F}}_{A B} = 1.35 N a l o n g \vec{A B}$

${\vec{F}}_{A C} = \frac{1}{4 {πε}_{\circ}} \frac{q_{A} q_{C}}{{r_{A C}}^{2}} a l o n g \vec{A C}$

${\vec{F}}_{A C} = 9 \times 10^{9} \times \frac{(2 \times 10^{- 6}) (3 \times 10^{- 6})}{{(0.20)}^{2}} a l o n g \vec{A C}$

${\vec{F}}_{A C} = 1.35 N a l o n g \vec{A C}$

${\vec{F}}_{A M} = \frac{1}{4 {πε}_{\circ}} \frac{q_{A} q_{M}}{{r_{A M}}^{2}} a l o n g \vec{M A}$

${\vec{F}}_{A M} = 9 \times 10^{9} \times \frac{(2 \times 10^{- 6}) (q_{M})}{{(\sqrt{3} \times 10^{- 1})}^{2}} a l o n g \vec{M A}$

${\vec{F}}_{A M} = 6 \times 10^{5} q_{M} N a l o n g \vec{M A}$

For equilibrium of charge $q_{A}$ : the resultant of ${\vec{F}}_{A B}$ and ${\vec{F}}_{A C}$ must be equal and opposite to ${\vec{F}}_{A M}$ .

i.e, $F_{A B} \cos 30^{\circ} + F_{A C} \cos 30^{\circ} = F_{A M}$

$1.35 \times \frac{\sqrt{3}}{2} + 1.35 \times \frac{\sqrt{3}}{2} = 6 \times 10^{5} q_{M} \Rightarrow q_{M} = \frac{1.35 \sqrt{3}}{6 \times 10^{5}} = 0.225 \sqrt{3} \times 10^{- 5} = 2.25 \sqrt{3} μ C$

Important Questions on Electric Charges and Fields

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Three point charges of +2μC,-3μC and -3μC are kept at the vertices A, B and C respectively of an equilateral triangle of side 20 cm as shown in figure. What should be the sign and magnitude of the charge to be placed at the mid-point M of side BC so that the charge at A remains in equilibrium?

Important Questions on Electric Charges and Fields

Learn and Prepare for any exam you want !