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The electric field at 2R from the centr...

The electric field at 2R from the centre of a uniformly charged non - conducting sphere of rarius R is E. The electric field at a distance `( R )/(2)` from the centre will be

A

Zero

B

2E

C

4E

D

16E

Text Solution

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The correct Answer is:
To solve the problem, we need to find the electric field at a distance of \( \frac{R}{2} \) from the center of a uniformly charged non-conducting sphere, given that the electric field at a distance of \( 2R \) from the center is \( E \). ### Step-by-Step Solution: 1. **Understand the Electric Field Outside the Sphere**: For a uniformly charged non-conducting sphere, the electric field outside the sphere (at a distance greater than the radius \( R \)) behaves like that of a point charge. The electric field \( E \) at a distance \( r \) from the center (where \( r > R \)) is given by: \[ E = \frac{KQ}{r^2} \] where \( K \) is Coulomb's constant and \( Q \) is the total charge of the sphere. 2. **Electric Field at Distance \( 2R \)**: We are given that the electric field at a distance \( 2R \) from the center is \( E \). Thus, we can write: \[ E = \frac{KQ}{(2R)^2} = \frac{KQ}{4R^2} \] 3. **Electric Field Inside the Sphere**: For points inside the uniformly charged non-conducting sphere (where \( r < R \)), the electric field \( E' \) is given by: \[ E' = \frac{KQ \cdot r}{R^3} \] where \( r \) is the distance from the center of the sphere. 4. **Calculate Electric Field at Distance \( \frac{R}{2} \)**: Now we need to find the electric field at a distance \( \frac{R}{2} \) from the center. We substitute \( r = \frac{R}{2} \) into the formula for the electric field inside the sphere: \[ E' = \frac{KQ \cdot \left(\frac{R}{2}\right)}{R^3} = \frac{KQ}{2R^2} \] 5. **Relate \( E' \) to \( E \)**: We already know that \( E = \frac{KQ}{4R^2} \). Now, we can express \( E' \) in terms of \( E \): \[ E' = \frac{KQ}{2R^2} = 2 \cdot \frac{KQ}{4R^2} = 2E \] ### Final Answer: Thus, the electric field at a distance \( \frac{R}{2} \) from the center of the sphere is: \[ E' = 2E \]
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Knowledge Check

  • The electric field at a distance (3R)/2 from the centre of a charged conducting spherical shell of radius R is E. The electric field at a distance R/2 from the centre of the sphere is :

    A
    `E`
    B
    `3E/2`
    C
    `E/2`
    D
    Zero
  • The electric field at a distance (3R)/2 from the centre of a charged conducting spherical shell of radiu R is E. The electric field at a distance R/2 from the centre of the sphere is:

    A
    zero
    B
    E
    C
    `E/2`
    D
    `E/3`
  • The electric field at a distance 3R//2 from the centre of a charge conducting spherical shell of radius R is E . The electric field at a distance R//2 from the centre of the sphere is

    A
    `(E )/(2)`
    B
    zero
    C
    `E`
    D
    `(E )/(4)`
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