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Let's consider one such element at a distance x from the center of the circle. The magnitude of the electric field dE due to this charge element can be given by Coulomb's Law:
dE = (k * dq) / r^2
Where:
- dE is the electric field due to a small charge element,
- k is the Coulomb's constant (k = 1 / (4πε₀), where ε₀ is the permittivity of free space),
- dq is the charge of the small element,
- r is the distance from the element to the center of the circle.
Now, since the charge is spread uniformly along the rod, we can express dq in terms of the total charge q and the length of the rod L:
dq = (q / L) * dx
Where:
- q is the total charge,
- L is the length of the rod,
- dx is the infinitesimally small length element.
Using the geometry of the problem, we can relate dx to the angle subtended at the center of the circle. Since the arc length of the rod is a * θ (where θ is the angle subtended), we can write:
dx = a * dθ
Now we can substitute dq and dx into the expression for dE:
dE = (k * dq) / r^2
= (k * (q / L) * dx) / r^2
= (k * (q / L) * (a * dθ)) / r^2
To find the total electric field E at the center of the circle, we integrate dE over the entire length of the rod. However, since the electric field due to all elements is along the same line, the direction of the electric field vectors will be the same and we can consider only the magnitudes:
E = ∫dE
= ∫[k * (q / L) * (a * dθ) / r^2]
Now, to express r in terms of a, L, and θ, we can use the geometry of the problem. The radius of the circle is a, and the distance from the center to the rod element (r) is given by:
r = a * sin(θ/2)
Substituting, we have:
E = ∫[k * (q / L) * (a * dθ) / (a * sin(θ/2))^2]
= (k * q / L) * ∫[dθ / (sin(θ/2))^2]
Integrating this expression will give us the final answer for the electric field at the center of the circle, in terms of a, q, and θ. The limits of integration will depend on the angle subtended by the rod.
Unfortunately, the integration involved in this problem is quite complex and cannot be solved analytically in a simple form. Therefore, to find a specific numerical value for the electric field, it would be necessary to perform numerical integration using appropriate computational methods.