A thin circular ring, of radius 20cm, is charged, with a uniform charge density(rho). If a small section, of 1cm length, is removed from the ring. Calculate the electric field intensity at the center of the ring.
To calculate the electric field intensity at the center of the ring, we can use the principle of superposition.
First, let's assume the charge density on the ring is defined as ρ (rho) coulombs per meter.
The electric field intensity due to a small element on the ring can be calculated using Coulomb's law. The magnitude of the electric field intensity (E) at the center of the ring due to a small section of length dl on the ring is given by:
dE = (k * dq) / r²
where k is the Coulomb's constant (approximately 9 × 10^9 N m²/C²), dq is the charge of the small section of length dl, and r is the radius of the ring.
To find dq, we need to calculate the charge for the small section of length 1 cm (0.01 m). The charge (dq) for a small element on the ring can be calculated using the charge density (ρ) and the length of the element (dl).
dq = ρ * dl
Since the ring is removed at one section of 1 cm length, we need to consider the charge for the remaining part of the ring, which is (Circumference - 1 cm).
The circumference of the ring is given by 2πr, where r is the radius. So, the charge (Q) for the remaining part of the ring can be calculated as:
Q = ρ * (2πr - 0.01)
Now, we can calculate the electric field intensity at the center due to the remaining charged ring. Considering symmetry, the contributions of electric field intensity from all the small sections of the ring cancel out, except for the diametrically opposite sections. Therefore, we can multiply the electric field intensity due to a small section by the number of such opposite sections, which is equal to half the total number of sections.
Using symmetry, we can write:
E = (dE * N) / 2
where N is the total number of sections on the ring.
Substituting the values, we get:
E = k * (Q / (2r)) * (1 / r²) * (2 / 2)
Simplifying further:
E = (kQ) / (4πr³)
Finally, substituting the given values (ρ = ? coulombs per meter and r = 20 cm = 0.2 m) into the equation, we can calculate the electric field intensity at the center of the ring.