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Question

Two identical charged spheres suspended from a common point by two massless strings of lengths $l$ , are initially at a distance ${\rm{d(d}}\, < \, < \,l)$  apart because of their mutual repulsion. The charges begin to leak from both the spheres at a constant rate. As a result, the spheres approach each other with a velocity  $\upsilon .\,{\rm{Then}}\,\upsilon $ varies as a function of the distance $x$ between the spheres, as

SUBHANI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: C
## CONCEPT FRAMEWORK
**Primary Concept**: Relationship between electrostatic force, equilibrium, and velocity
**Related Topics**: Coulomb's law, small angle approximation, differentiation, Newton's second law
**Prerequisites**: Basic understanding of electrostatics, trigonometry, and calculus
## SOLUTION PATHWAY
**Given Information**:
* Two identical charged spheres suspended by strings of length $l$.
* Initial separation: $d$ ($d << l$)
* Charges leak at a constant rate.
* Spheres approach each other with velocity $\upsilon$
* Distance between spheres is $x$
**Method & Steps**:
1. **Analyze forces at equilibrium**: At any distance $x$, the electrostatic force $F_e$ is balanced by the horizontal component of tension $T$ in the string. Using small angle approximation: $F_e = T \sin	heta \approx T	heta \approx T\frac{x}{2l}$. Since $T \approx mg$, we have $F_e \approx mg\frac{x}{2l}$.
Reasoning: Force balance at equilibrium
Formula/Rule: Small angle approximation, force equilibrium
2. **Express electrostatic force**: The electrostatic force between two charged spheres is given by Coulomb's Law: $F_e = k\frac{q^2}{x^2}$ where $k$ is Coulomb's constant and q is the charge on each sphere.
Reasoning: Coulomb's Law
Formula/Rule: $F_e = k\frac{q^2}{x^2}$
3. **Relate force and distance**: Equating the expressions for electrostatic force from steps 1 and 2, we get $k\frac{q^2}{x^2} \approx mg\frac{x}{2l}$. This gives us $q^2 \propto x^3$.
Working: Equating forces
Key Point: The charge is decreasing, so $q$ is a function of time
4. **Differentiate with respect to time**: Since charge leaks at a constant rate, $\frac{dq}{dt} = C$ (constant). Differentiating $q^2 \propto x^3$ with respect to time, we have $2q\frac{dq}{dt} \propto 3x^2\frac{dx}{dt}$. Since $\frac{dq}{dt}$ is constant, we get $q \propto x^2 \frac{dx}{dt}$.
Working: Differentiation with respect to time.
Key Point: The rate of change of charge is constant.
5. **Substitute q**: From step 3, $q \propto x^{3/2}$, so we get  $x^{3/2} \propto x^2 \frac{dx}{dt}$. Simplifying, $\frac{dx}{dt} \propto x^{-1/2}$. Since $\upsilon = \frac{dx}{dt}$, we have $\upsilon \propto x^{-1/2}$
Result: Final relationship between velocity and distance.
Verification: Dimensional analysis
## OPTION ANALYSIS
[A] :x: Incorrect relationship. The velocity is inversely proportional to the square root of distance
[B] :x: Incorrect relationship. The velocity is inversely proportional to the square root of distance
[C] ✓: Correct. The velocity is inversely proportional to the square root of distance
[D] :x: Incorrect relationship. The velocity is inversely proportional to the square root of distance
## LEARNING AIDS
**Common Mistakes**:
1. **Incorrect small angle approximation**: Using $\sin	heta \approx 	heta$ is valid for small angles. Make sure to apply it correctly.
2. **Not differentiating with respect to time**: The charges are changing with time so it is important to differentiate with respect to time to find the relationship between velocity and distance
**Quick Tips**:
1. **Use small angle approximation**: simplifies the analysis when angles are small
2. **Relate forces to distance**: relate the electrostatic force to the distance between the charges

SUBHANI · Answer

${\rm{\upsilon }}\,{\rm{\alpha }}\,{{\rm{x}}^{\frac{{\rm{1}}}{{\rm{2}}}}}$

SUBHANI · Answer

${\rm{\upsilon }}\,{\rm{\alpha }}\,{\rm{x}}$

SUBHANI · Answer

The correct Option is "${\rm{\upsilon }}\,{\rm{\alpha }}\,{{\rm{x}}^{ - \frac{{\rm{1}}}{{\rm{2}}}}}$"

SUBHANI · Answer

${\rm{\upsilon }}\,{\rm{\alpha }}\,{{\rm{x}}^{ - {\rm{1}}}}$

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