The electric field on the axis of a uniformly charged ring has magnitude 370kN/C at a point 6.2cm from the ring center. The magnitude 17cm from the center is 150kN/C ; in both cases the field points away from the ring.Find the ring's radius.
To find the radius of the uniformly charged ring, we can use the formula for the electric field on the axis of a uniformly charged ring:
E = (k * Q * x) / ((x^2 + R^2)^(3/2))
Where:
E is the magnitude of the electric field
k is the Coulomb's constant (approximately 9 x 10^9 Nm^2/C^2)
Q is the total charge of the ring
x is the distance from the center of the ring along the axis
R is the radius of the ring
We are given the following information:
E1 = 370 kN/C (at a distance of 6.2 cm)
E2 = 150 kN/C (at a distance of 17 cm)
We can use these values to set up two equations:
E1 = (k * Q * x1) / ((x1^2 + R^2)^(3/2))
E2 = (k * Q * x2) / ((x2^2 + R^2)^(3/2))
Now, let's solve these equations to find the values of x1, x2, and R.
First, let's convert the distances to meters:
x1 = 6.2 cm = 0.062 m
x2 = 17 cm = 0.17 m
Substituting these values into the equations:
370 kN/C = (k * Q * 0.062) / ((0.062^2 + R^2)^(3/2))
150 kN/C = (k * Q * 0.17) / ((0.17^2 + R^2)^(3/2))
Now, we can divide the second equation by the first equation to eliminate Q:
150 kN/C / 370 kN/C = (k * Q * 0.17) / ((0.17^2 + R^2)^(3/2)) / ((k * Q * 0.062) / ((0.062^2 + R^2)^(3/2)))
0.4054 = (0.17 / ((0.17^2 + R^2)^(3/2))) / (0.062 / ((0.062^2 + R^2)^(3/2)))
Now, let's cross-multiply and simplify the equation:
(0.17^2 + R^2)^(3/2) = (0.062^2 + R^2)^(3/2) * 0.4054
Square both sides of the equation:
(0.17^2 + R^2)^3 = (0.062^2 + R^2)^3 * 0.4054^2
Expand and simplify:
0.4913 + 2 * 0.4913 * R^2 + R^4 = 0.0624 + 2 * 0.0624 * R^2 + R^4 * 0.165
Rearrange the equation:
0.4913 + 2 * 0.4913 * R^2 - 0.0624 - 2 * 0.0624 * R^2 = R^4 * 0.165 - R^4
Combine like terms:
0.4286 = 0.1336 * R^4
Now, let's solve for R^4:
R^4 = 0.4286 / 0.1336
R^4 = 3.2088
Taking the fourth root of both sides:
R = (3.2088)^(1/4)
R ≈ 1.27
Therefore, the radius of the uniformly charged ring is approximately 1.27 meters.