Question

A net charge of 0.2 pC (picocoulomb) is uniformly distributed on a circular hoop of radius y=0.3 m. a) Determine the potential and the electric field strength (E= -dV/dz) at the center of the hoop (z=0) and at a point on the axis of the hoop 0.4 m from the center. (10 points) b) Where on the axis is the field strength a maximum? (5 points) (Hint: to determine maximum of E, equate dE/dz = 0, and find the value of z). What is the maximum field strength at this point?

Answer 1

a)

electric potential by a hoop of radius "y" carrying charge "Q" ,at a distance "z" from the center along the axis is given as

V = K Q/sqrt(y² + z²)

when z = 0

V = K Q/sqrt(y² + 0²) = K Q/y = (9 x 10⁹) (0.2 x 10^-12)/(0.3) = 0.006 Volts

and

E = kQz/(y² + z²)^3/2

at z = 0

E = kQ(0)/(y² + z²)^3/2 = 0 N/C

b)

E = kQz/(y² + z²)^3/2

dE/dz = kQ ((dz/dz) (y² + z²)^3/2 - z (3/2) (y² + z²)^1/2 (2z)) = 0

(y² + z²)^3/2 = 2z² (1.5) (y² + z²)^1/2  

(y² + z²) = 3 z²

2z² = y²

z = y/sqrt(2)

E_max = kQz/(y² + z²)^3/2 = kQ(y/sqrt(2))/(y² + (y/sqrt(2))²)^3/2

E_max = kQ(y/sqrt(2))/(1.5 y²)^3/2 = (9 x 10⁹) (0.2 x 10^-12) (0.707) (0.3)/(1.5 (0.3)²)^3/2? = 7.7 x 10^-3 N/c