Consider an infinitely long cylinder of radius R , placed along the z -axis, which carries a static charge density r  kr , where r is the perpendicular distance from the axis of the cylinder and k is a constant. The electrostatic potential  r inside the cylinder is proportional t?

Question

Answer

Infinitely Long Cylinder with Static Charge Density

In this problem, we are given an infinitely long cylinder of radius R placed along the z-axis. The cylinder carries a static charge density ρ(r) = kr, where r is the perpendicular distance from the axis of the cylinder and k is a constant. We need to determine the electrostatic potential ϕ(r) inside the cylinder and show that it is proportional to r^2.

Understanding the Problem

To solve this problem, let's first understand the given scenario and the concepts involved:

The cylinder has a charge distribution that depends on the perpendicular distance from its axis.

The electrostatic potential inside the cylinder is the scalar field that describes the potential energy of a unit positive charge placed at a certain point inside the cylinder.

We need to find an expression for the electrostatic potential ϕ(r) inside the cylinder based on the given charge density ρ(r).

Solution

To find the electrostatic potential inside the cylinder, we can use the Gauss's law and the concept of electric potential:

Gauss's Law: The electric flux through a closed surface is equal to the total charge enclosed by that surface divided by the permittivity of free space.

Electric Potential: The potential difference between two points in an electric field is equal to the work done per unit charge in moving a positive test charge from one point to another.

Applying Gauss's Law

Since the cylinder is infinitely long, we can consider a cylindrical Gaussian surface of radius r and length L inside the cylinder.

The electric field is radial and its magnitude is given by E = k*r/ε, where ε is the permittivity of free space.

The flux through the Gaussian surface is Φ = E * A, where A is the area of the curved surface of the cylinder.

The charge enclosed by the Gaussian surface is Q = ∫ρ(r) * dV, where dV is the volume element inside the cylinder.

Calculating the Electric Field

The electric field inside the cylinder is given by E = k*r/ε.

The electric potential difference between two points inside the cylinder is given by Δϕ = -∫E * dr.

Determining the Electrostatic Potential

Integrating the electric field expression, we get Δϕ = -∫(k*r/ε) * dr.

Simplifying the integral, we find Δϕ = -k/(2*ε) * r^2 + C, where C is the constant of integration.

The electrostatic potential at a point inside the cylinder is ϕ(r) = -k/(2*ε) * r^2 + C.

Conclusion

From the above analysis, we have determined the electrostatic potential

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