Eshan Sir …

But in our problem, sphere given.

You told E

electric field is dependent on r in sphere ,

But here E independent on r, how sir..please once explain.

It's a common point of confusion when discussing electric fields, especially in relation to spheres. Let's break it down step by step to clarify how the electric field behaves in different scenarios involving spheres.

Understanding Electric Fields Around Spheres

The electric field (E) generated by a charged object can vary depending on the shape of the object and the distribution of charge. For spheres, we typically consider two cases: when the charge is uniformly distributed on the surface and when it is distributed throughout the volume.

Case 1: Charged Sphere with Uniform Surface Charge

When we have a sphere with a uniform charge distributed over its surface, the electric field outside the sphere behaves as if all the charge were concentrated at a point at the center of the sphere. This means that the electric field (E) at a distance (r) from the center of the sphere is given by:

• E = k * Q / r²

Here, k is Coulomb's constant, and Q is the total charge on the sphere. As you can see, in this case, the electric field does depend on the distance (r) from the center of the sphere. The farther away you are from the sphere, the weaker the electric field becomes, following an inverse square law.

Case 2: Charged Sphere with Uniform Volume Charge

Now, let’s consider a solid sphere with a uniform charge distributed throughout its volume. Inside this sphere, the electric field behaves differently. According to Gauss's Law, the electric field inside a uniformly charged sphere increases linearly with distance from the center:

• E = (1/4πε₀) * (Q * r) / R³ for r < R

Where R is the radius of the sphere and ε₀ is the permittivity of free space. Here, r is the distance from the center of the sphere, and the electric field increases as you move away from the center, up to the surface.

Outside the Sphere

Once you are outside the sphere (r > R), the electric field again follows the inverse square law, just like in the first case:

• E = k * Q / r²

This means that outside the sphere, the electric field is indeed dependent on the distance from the center, just as you initially understood.

Why the Confusion?

The confusion often arises when discussing the electric field at points inside a charged sphere versus points outside it. Inside a uniformly charged solid sphere, the electric field is not constant; it varies with distance from the center. However, if you are considering a point at the surface or outside the sphere, the electric field does depend on the distance from the center.

Visualizing the Concept

Think of it like this: if you are standing inside a balloon filled with air (representing the charged sphere), you would feel the pressure (electric field) increasing as you move towards the surface. Once you step outside the balloon, the pressure (electric field) decreases as you move further away.

In summary, the electric field around a sphere can be dependent on distance in certain contexts, particularly when considering the position relative to the sphere's surface. Understanding these distinctions is key to grasping the behavior of electric fields in relation to spherical charge distributions.