The total charge on a uniformly charged ring with diameter 26.0 cm is −57.3 µC. What is the magnitude of the electric field along the ring's axis at the following distances from its center?
(a) 2.60 cmN/C, (b) 13.0 cmN/C, (c) 26.0 cm N/C, (d) 2.08 m N/C
a = radius = 0.13 meter
x = distance from center along that center line
is charge in coulombs
I do not know if you have been given the formula, it is:
Ex = (1/ [4pi eo] ) Q x / (x^2+a^2)^1.5
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If you need the derivation
dQ at spot along ring
let k = (1/ [4pi eo] ) for now
dE = k dQ /(x^2+a^2)
x component of that (the y components cancel of course)
dEx =dE cos angle = k dQ /(x^2+a^2) * x/sqrt(x^2+a^2)
= k dQ [ x/ (x^2+a^2)^1.5 ]
integrate dQ at our chosen constant x
Ex = k Q [ x/ (x^2+a^2)^1.5 ]
To find the magnitude of the electric field along the ring's axis at different distances from its center, we can use the formula for the electric field due to a uniformly charged ring.
The formula for the electric field due to a uniformly charged ring at a point along its axis is given by:
E = k * (Q * z) / (2 * R^3)
Where:
- E is the electric field
- k is the electrostatic constant (k = 9.0 x 10^9 N·m^2/C^2)
- Q is the total charge on the ring
- z is the distance along the axis from the center of the ring
- R is the radius of the ring
Let's calculate the magnitude of the electric field at each given distance:
(a) Distance = 2.60 cm = 0.0260 m
Plug the values into the formula:
E = (9.0 x 10^9 N·m^2/C^2) * ((-57.3 x 10^-6 C) * 0.0260 m) / (2 * (0.260 m)^3)
Solve for E:
E = -27.5 N/C
(b) Distance = 13.0 cm = 0.13 m
Plug the values into the formula:
E = (9.0 x 10^9 N·m^2/C^2) * ((-57.3 x 10^-6 C) * 0.13 m) / (2 * (0.260 m)^3)
Solve for E:
E = -550 N/C
(c) Distance = 26.0 cm = 0.26 m
Plug the values into the formula:
E = (9.0 x 10^9 N·m^2/C^2) * ((-57.3 x 10^-6 C) * 0.26 m) / (2 * (0.260 m)^3)
Solve for E:
E = -2.20 x 10^3 N/C
(d) Distance = 2.08 m
Plug the values into the formula:
E = (9.0 x 10^9 N·m^2/C^2) * ((-57.3 x 10^-6 C) * 2.08 m) / (2 * (0.260 m)^3)
Solve for E:
E = -7.55 x 10^6 N/C
Therefore, the magnitude of the electric field along the ring's axis at the given distances are:
(a) 27.5 N/C
(b) 550 N/C
(c) 2.20 x 10^3 N/C
(d) 7.55 x 10^6 N/C