SATHEE: Electrostatics 1 Question 24

Electrostatics 1 Question 24

25. A rigid insulated wire frame in the form of a right angled triangle $A B C$ , is set in a vertical plane as shown in figure. Two beads of equal masses $m$ each and carrying charges $q_{1}$ and $q_{2}$ are connected by a cord of length $l$ and can slide without friction on the wires.

Considering the case when the beads are stationary determine

(1978)

(a) (i) The angle $α$

(ii) The tension in the cord

(iii) The normal reaction on the beads

(b) If the cord is now cut what are the value of the charges for which the beads continue to remain stationary?

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Answer:

Correct Answer: 25. (a) (i) $α = 60^{\circ}$

(ii) $T = (\frac{1}{4 π ε_{0}}) \frac{q_{1} q_{2}}{l^{2}} + m g$

(iii) $N_{P} = \sqrt{3} m g, N_{Q} = m g$

(b) $q_{1} q_{2} = - (- 4 π ε_{0}) m g l^{2}$

Solution:

Tension and electrostatic force are in opposite direction and along the string. Now each bead is in equilibrium under three concurrent forces

(i) Normal reaction $(N)$

(ii) Weight $(m g)$ and

(iii) $T - F_{e}$ , where $F_{e} = \frac{1}{4 π ε_{0}} \cdot \frac{q_{1} q_{2}}{l^{2}}$

Applying Lami’s theorem for both beads.

$\begin{aligned} \frac{N_{1}}{\sin (90^{\circ} - α)} = \frac{m g}{\cos α} = \frac{T - F_{e}}{\cos 60^{\circ}} \\ \frac{N_{2}}{\sin (60^{\circ} + α)} = \frac{m g}{\sin α} = \frac{T - F_{e}}{\cos 30^{\circ}} \end{aligned}$

Dividing Eq. (i) by Eq. (ii), we have

$\begin{aligned} \tan α & = \frac{\cos 30^{\circ}}{\cos 60^{\circ}} = \sqrt{3} \Rightarrow α = 60^{\circ} \\ T & = F_{e} + m g = \frac{1}{4 π ε_{0}} \cdot \frac{q_{1} q_{2}}{l^{2}} + m g \\ N_{1} & = \sqrt{3} m g and N_{2} = m g \end{aligned}$

From Eq. (iii) $T = 0$ when string is cut.

$or q_{1} q_{2} = - (4 π ε_{0}) m g l^{2}$

Electrostatics 1 Question 24

25. A rigid insulated wire frame in the form of a right angled triangle $A B C$ , is set in a vertical plane as shown in figure. Two beads of equal masses $m$ each and carrying charges $q_{1}$ and $q_{2}$ are connected by a cord of length $l$ and can slide without friction on the wires.

Answer:

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Electrostatics 1 Question 24

25. A rigid insulated wire frame in the form of a right angled triangle ABC, is set in a vertical plane as shown in figure. Two beads of equal masses m each and carrying charges q1 and q2 are connected by a cord of length l and can slide without friction on the wires.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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25. A rigid insulated wire frame in the form of a right angled triangle $A B C$ , is set in a vertical plane as shown in figure. Two beads of equal masses $m$ each and carrying charges $q_{1}$ and $q_{2}$ are connected by a cord of length $l$ and can slide without friction on the wires.