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The potential of the field inside a char...

The potential of the field inside a charged sphere depends upon the distance from its centre to the point under consideration in the following way:
`V=ar^n+b`, where a and b are constants
If the charge is uniformly distributed within the sphere, then the value of n is

A

2

B

`-2`

C

1

D

`-1`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to determine the value of \( n \) in the equation for the potential \( V \) inside a uniformly charged sphere, given by: \[ V = ar^n + b \] where \( a \) and \( b \) are constants. ### Step-by-Step Solution: 1. **Understanding the Potential Inside a Charged Sphere**: The potential \( V \) at a distance \( r \) from the center of a uniformly charged sphere of radius \( R \) and total charge \( Q \) is given by the formula: \[ V = k \frac{Q}{2R} (3R^2 - r^2) \] where \( k \) is Coulomb's constant. 2. **Identifying the Form of the Potential**: We can rewrite the potential equation: \[ V = \frac{3kQ}{2R} R^2 - \frac{kQ}{2R} r^2 \] This shows that the potential is a linear combination of a constant term and a term that depends on \( r^2 \). 3. **Comparing with the Given Form**: The given form of the potential is: \[ V = ar^n + b \] From our derived expression, we can see that: - The term \( -\frac{kQ}{2R} r^2 \) corresponds to \( ar^n \). - The constant term \( \frac{3kQ}{2R} R^2 \) corresponds to \( b \). 4. **Identifying the Value of \( n \)**: Since the term involving \( r \) in our potential expression is \( -\frac{kQ}{2R} r^2 \), we can identify: \[ a = -\frac{kQ}{2R} \quad \text{and} \quad n = 2 \] 5. **Conclusion**: Therefore, the value of \( n \) is: \[ n = 2 \] ### Final Answer: The value of \( n \) is \( 2 \). ---
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