Two tiny spheres, each having mass m kg and charge q coulomb, are suspended from a point by insulating threads each 1 metre long but negligible mass. When the system is in equilibrium, each string makes an angle θ with the vertical. Prove that q² = (4 mg l² sin² θ tan θ) $4 {πε}_{0} .$

Solution

Consider the equilibrium of sphere A.
Following forces act on the sphere A.
(i) Force F of repulsion on A due to B.
(ii) Weight mg acting vertically downwards.
(iii) Tension T in the string towards the point of suspension O.

Resolving the tension T into two rectangular components:

$T \cos θ and T \sin θ$

For equilibrium of A,
$T \sin θ = F$
   $= \frac{1}{4 {πε}_{0}} \frac{q^{2}}{{AB}^{2}}$
and
   $T \cos θ = mg$

Dividing,
   $\tan θ = \frac{1}{4 {πε}_{0}} \frac{q^{2}}{{AB}^{2} mg}$

But $AB = 2 AC$
   $= 2 l \sin θ$

$∴$ $\tan θ = \frac{1}{4 {πε}_{0}} \frac{q^{2}}{(2 l \sin θ)^{2} mg}$

$q^{2} = (4 m g l^{2} s i n^{2} θ t a n θ) 4 π ε_{0} .$

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Electric Charges And Fields

Two tiny spheres, each having mass m kg and charge q coulomb, are suspended from a point by insulating threads each 1 metre long but negligible mass. When the system is in equilibrium, each string makes an angle θ with the vertical. Prove that q² = (4 mg l² sin² θ tan θ) $4 {πε}_{0} .$

Some More Questions From Electric Charges and Fields Chapter

a) An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?
b) Explain why two field lines never cross each other at any point?

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Electric Charges And Fields

Two tiny spheres, each having mass m kg and charge q coulomb, are suspended from a point by insulating threads each 1 metre long but negligible mass. When the system is in equilibrium, each string makes an angle θ with the vertical. Prove that q2 = (4 mg l2 sin2 θ tan θ) 4πε0.

Some More Questions From Electric Charges and Fields Chapter

a) An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?b) Explain why two field lines never cross each other at any point?

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Two tiny spheres, each having mass m kg and charge q coulomb, are suspended from a point by insulating threads each 1 metre long but negligible mass. When the system is in equilibrium, each string makes an angle θ with the vertical. Prove that q² = (4 mg l² sin² θ tan θ) $4 {πε}_{0} .$

a) An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not?
b) Explain why two field lines never cross each other at any point?