A thin non-conducting rod is bent to form the arc of a circle of radius a and subtends an angle at the center of the circle. A total charge q is spread uniformly along its length. Find the electric field at the center of the circle in term of, a, q, and angle.
To find the electric field at the center of the circle, we can divide the thin non-conducting rod into small elements and calculate the electric field contribution from each element. Then, by considering the symmetry of the problem, we can find the net electric field at the center of the circle.
Let's consider a small element of length ΔL on the rod. The charge on this element is given by Δq = q * (ΔL / L), where L is the total length of the rod.
The electric field dE at the center of the circle due to this small element can be found using Coulomb's law:
dE = k * (Δq / r^2)
Where k is the electrostatic constant (k = 1 / (4πε₀), where ε₀ is the vacuum permittivity) and r is the distance from the small element to the center of the circle.
Since the rod is bent to form the arc of a circle, the distance r is constant and equal to the radius of the circle, a.
So, dE = k * (Δq / a^2)
To find the net electric field at the center, we need to integrate the contributions from all small elements of the rod. The total electric field at the center, E, is given by:
E = ∫ dE
Integrating over the length of the rod will give us the total electric field.
E = ∫[k * (Δq / a^2)] = k / a^2 ∫Δq
Now, we substitute Δq with q * (ΔL / L):
E = k / a^2 ∫[q * (ΔL / L)]
Since the charge q is spread uniformly along the length of the rod, the integral can be simplified:
E = k / a^2 * (q / L) * ∫dL
The integral ∫dL represents the total length of the rod: L. So the expression simplifies to:
E = k * q / (a^2 * L)
Finally, we can substitute the given angle in terms of length and radius using the relation s = rθ, where s is the arc length on the circle subtended by the angle θ:
L = aθ
Substituting this into the expression for the electric field:
E = k * q / (a^2 * L) = k * q / (a^2 * aθ) = k * q / (a^3θ)
Therefore, the electric field at the center of the circle in terms of a, q, and the angle θ is:
E = k * q / (a^3θ)