A conducting solid sphere of radius 20.0 cm is located with its center at the origin of a three-dimensional coordinate system. A charge of 0.343 nC is applied to the sphere. (a) What is the magnitude of the electric field at point (x,y,z) = (21.3 cm,5.43 cm,0 cm)? N/C (b) What is the angle of this electric field with respect to the x-axis at this point? (Please give your answer in degrees.) (c) What is the magnitude of the electric field at point (x,y,z) = (4.09 cm,5.43 cm,0 cm)? N/C
Question
A conducting solid sphere of radius 20.0 cm is located with its center at the origin of a three-dimensional coordinate system. A charge of 0.343 nC is applied to the sphere. (a) What is the magnitude of the electric field at point (x,y,z) = (21.3 cm,5.43 cm,0 cm)? N/C (b) What is the angle of this electric field with respect to the x-axis at this point? (Please give your answer in degrees.) (c) What is the magnitude of the electric field at point (x,y,z) = (4.09 cm,5.43 cm,0 cm)? N/C
Solution
(a) The electric field E outside a charged sphere of radius R and total charge Q is given by Coulomb's law:
E = kQ/r^2
where k is Coulomb's constant (8.99 x 10^9 N m^2/C^2), Q is the total charge, and r is the distance from the center of the sphere.
First, we need to convert the charge from nC to C: 0.343 nC = 0.343 x 10^-9 C.
Next, we need to find the distance r from the origin to the point (21.3 cm, 5.43 cm, 0 cm). This is given by the Pythagorean theorem:
r = sqrt(x^2 + y^2 + z^2) = sqrt((21.3 cm)^2 + (5.43 cm)^2 + (0 cm)^2) = 22.0 cm = 0.22 m.
Now we can calculate the electric field:
E = (8.99 x 10^9 N m^2/C^2) * (0.343 x 10^-9 C) / (0.22 m)^2 = 6.00 x 10^4 N/C.
(b) The angle θ of the electric field with respect to the x-axis is given by:
θ = arctan(y/x) = arctan(5.43 cm / 21.3 cm) = 14.4 degrees.
(c) To find the electric field at the point (4.09 cm, 5.43 cm, 0 cm), we first find the distance r from the origin to this point:
r = sqrt((4.09 cm)^2 + (5.43 cm)^2 + (0 cm)^2) = 6.79 cm = 0.068 m.
Since this point is inside the sphere (r < R), the electric field is zero:
E = 0 N/C.
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