Two identical balls each of mass m and charge q are suspended by two string of length ℓ from common

Two identical balls each of mass m and charge q are suspended by two string of length ℓ from common point at equilibrium angle between strings is 90°. The charge q is :-
(a) (4πε0 mgℓ^2)^1/2 (b) (8πε0 mgℓ^2)^1/2
(c) (πε0 mgℓ^2)^1/2 (d) None
*Charge on Suspended Identical Balls - Equilibrium Analysis* ⚡️
In this video, we solve a fascinating problem involving two identical balls, each with mass *m* and charge *q**, suspended by strings of length **ℓ**. At equilibrium, the angle between the strings is **90°**. Our goal is to determine the expression for the charge **q* that keeps the system in balance.
*Problem Overview* 💡:
Two identical balls, each having mass *m* and charge *q**, are suspended by two strings of length **ℓ**. The balls are in equilibrium, with the angle between the strings being **90°**. We need to determine the expression for the charge **q* that maintains this equilibrium.
*Key Concepts* 🔍:
**Forces Acting on the Balls**:
**Gravitational Force**: Each ball experiences a downward force due to gravity, **mg**.
**Electric Force**: The repulsive force between the two charges.
**Tension in the Strings**: The tension in each string balances both the gravitational and electric forces.
**Equilibrium Condition**:
At equilibrium, the forces acting on each ball must balance in both horizontal and vertical directions.
*Solution Steps* 🧠:
1. **Vertical Force Balance**:
The vertical component of the tension in the string must balance the gravitational force (**mg**).
2. **Horizontal Force Balance**:
The horizontal component of the tension must balance the repulsive electric force between the charges.
3. **Using Coulomb's Law**:
The electric force between the charges is given by:
\[
F = \frac{1}{4 \pi \varepsilon_0} \cdot \frac{q^2}{r^2}
\]
where *r = ℓ√2* is the distance between the charges and *\( \varepsilon_0 \)* is the permittivity of free space.
4. **Solving for Charge**:
By solving the force balance equations, we derive the expression for the charge *q* as:
\[
q = \sqrt{8 \pi \varepsilon_0 \cdot mg \cdot \ell^2}
\]
*Conclusion* 🔑:
The charge *q* required to keep the system in equilibrium is **\( (8 \pi \varepsilon_0 mg \ell^2)^{1/2} \)**, which corresponds to option **(b)**.
*Key Takeaways* 🔑:
This problem demonstrates the interaction between gravitational and electric forces in a system of charges.
The equilibrium condition requires balancing both horizontal and vertical forces.
Coulomb’s law is a key tool in determining the electric force between charges in such problems.
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