Question

A ring with radius r and a uniformly distributed total charge Q lies in the xy plane, centered at the originPart a What is the potential V(z) due to the ring on the z axis as a function of z? Express your answer in terms of Q ,r , z, and or ke = 1/4 phi e0 PART B What is the magnitude of the electric field E on the z axis as a function of z , for z > 0? Express your answer in terms of some or all of the quantities Q,z , r, and or ke = 1/4 phi e0

Answer 1

The electric potential V(z) due to a ring of charge on the z-axis is V(z) = ke * Q / sqrt(z^2 + r^2), and the magnitude of the electric field on the z-axis is E(z) = ke * Q * z / (z^2 + r^2)^(3/2).

Step-by-step explanation:

Part A: Electric Potential V(z) Due to a Ring of Charge

The electric potential V(z) at a distance z on the axis of a ring with radius r and total charge Q can be found using the concept of a point charge potential.

The potential is given by the expression:

V(z) = \( \frac{ke \cdot Q}{\sqrt{z^2 + r^2}} \)

Where ke is the Coulomb's constant ke = \( \frac{1}{4 \pi \varepsilon_0} \), Q is the total charge,

r is the radius of the ring,

And z is the distance along the z-axis from the center of the ring.

Part B: Electric Field E on the Z-Axis as a Function of Z

To find the magnitude of the electric field E at a point z on the axis of the ring, we differentiate the potential concerning z to obtain:

E(z) = \( \frac{ke \cdot Q \cdot z}{(z^2 + r^2)^{3/2}} \)

This provides us with the magnitude of the electric field due to the ring of charge at a point along the z-axis, for z > 0.

Answer 2

For a uniformly charged ring on the z-axis, the potential V(z) can be derived by integrating the contributions of individual charge elements. The corresponding electric field Ez is then obtained by taking the negative derivative of V(z) with respect to z.

The potential and electric field of a uniformly charged ring on the z-axis:

Part A: Potential V(z)

Divide the ring into small segments: Imagine the ring as being made up of many tiny charge elements, each with a small amount of charge dq. The total charge Q of the ring is the sum of the charges of all these elements.

Calculate the potential due to a single charge element: The potential at a point z due to a single charge element dq is given by:

dV = k * dq / sqrt(r^2 + z^2)

where k is Coulomb's constant (k = 1/4πε₀), r is the distance from the charge element to the point on the z-axis, and dq is the charge of the element.

Integrate over the entire ring: To find the potential due to the entire ring, we need to sum the potentials of all the charge elements. This can be done using integration:

V(z) = ∫ dV = k ∫ dq / sqrt(r^2 + z^2)

where the integral is taken over the entire ring.

Substitute for dq and r: The charge dq on each element is related to the total charge Q and the circumference of the ring (2πR) by:

dq = Q / (2πR)

The distance r from a charge element to a point on the z-axis is given by the Pythagorean theorem:

r^2 = R^2 + a^2

where a is the distance from the center of the ring to the point on the z-axis.

Solve the integral: Substituting for dq and r, and using the trigonometric substitution a = R tan(θ), we can solve the integral to get the potential V(z) as a function of z, Q, R, and k.

Part B: Electric Field E(z)

Relate potential and electric field: The electric field E is related to the potential V by the negative gradient of V:

E = -∇V

In this case, the electric field has only one non-zero component, Ez, which points along the z-axis.

Calculate Ez: Taking the negative derivative of V(z) with respect to z, we can find the magnitude of the electric field Ez as a function of z, Q, R, and k.