A non-conducting ring of radius R with a uniform charge density λ and a total charge Q is lying in the yz - plane, as shown in the figure below. Consider a point P, located at a distance x from the center of the ring along its axis of symmetry.
1) What is the magnitude (in N/C) of the electric field at P? Express your answer in terms of the following variables, if necessary, Q, R, x and the constant ϵ0 (enter pi for π, epsilon_0 for ϵ0). Do not use λ in your answer.
(x*Q)/(4*pi*epsilon_0*((R)^2+(x)^2)^(3/2))
To find the magnitude of the electric field at point P due to the non-conducting ring, we can use the electric field formula for a continuous charge distribution.
Step 1: Determine the electric field contribution from an infinitesimally small element of charge on the ring.
Consider an infinitesimally small element of charge on the ring, dQ, located at an angle θ from the positive x-axis. The charge on this small element can be expressed as dQ = λdθ, where λ is the linear charge density of the ring.
The electric field contribution from this small element of charge can be given by Coulomb’s law:
dE = k * (dQ) / r²
where k is the Coulomb's constant, dQ is the charge on the element, and r is the distance between the element and point P.
In this case, the distance between the small element and point P can be expressed as:
r = √(x² + R²)
Step 2: Integrate the electric field contribution for the complete ring.
To find the total electric field at point P, we need to integrate the electric field contribution from all the small elements around the ring.
E = ∫dE
To simplify the integration, it is convenient to express the charge distribution in terms of the angle θ, rather than x. The angle θ can be related to the distance x using trigonometry.
For a ring, the linear charge density λ is constant, so we can express it as:
λ = Q / (2πR)
where Q is the total charge on the ring and R is the radius of the ring.
Substituting λ = Q / (2πR) into the expression for dQ, we get:
dQ = (Q / 2πR)dθ
Substituting all the values into the electric field expression and integrating, we get:
E = ∫ dE = ∫ [k * (Q / 2πR) * dθ / (√(x² + R²))²]
Simplifying the integral and the expression, we get:
E = kQx / (4πϵ₀(x² + R²)^(3/2))
where ϵ₀ is the permittivity of free space.
So, the magnitude of the electric field at point P is given by:
E = kQx / (4πϵ₀(x² + R²)^(3/2))