A uniformly charged insulating rod of length 19.5 cm is bent into the shape of a semicircle as shown in the figure below. The rod has a total charge of −7.50 µC. A rectangular rod is bent into the shape of the left half of a circle centered about a point O. (a) Find the magnitude of the electric field at O, the center of the semicircle. (b) Find the direction of the electric field at O, the center of the semicircle.
A uniformly charged insulating rod of length 19.5 cm is bent into the shape of a semicircle as shown in the figure below. The rod has a total charge of −7.50 µC. A rectangular rod is bent into the shape of the left half of a circle centered about a point O. (a) Find the magnitude of the electric field at O, the center of the semicircle. (b) Find the direction of the electric field at O, the center of the semicircle.
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A uniformly charged insulating rod of length 19.5 cm is bent into the shape of a semicircle as shown in the figure below. The rod has a total charge of −7.50 µC.
A rectangular rod is bent into the shape of the left half of a circle centered about a point O.
(a) Find the magnitude of the electric field at O, the center of the semicircle.
(b) Find the direction of the electric field at O, the center of the semicircle.
(b) Find the direction of the electric field at O, the center of the semicircle.
Transcribed Image Text:(a) Let A be the charge per unit length and let r be the radius of the semicircular ring. From the diagram, we see that
dq = Arde.
The field created by the charge dq has magnitude
kda kArde
dE =
r2
r2
In component form, E, = 0 (from symmetry) and for the x component, we have
dE, = dE cos e.
Integrating dE, over all of the charge, we have the following.
kar cos 8
ka (T/2
Ex =
de =
cos e de
-n/2
dE, =
r2
2k
sin
%3D
-
-피/2
For the total charge, we have Q = Ae, where e is the length of the rod. The rod has been bent into a semicircle so ar = l. For the radius, we have r = e/n. Substituting these expressions for 2 and r, we
have the following.
2nk Q
Ex =
2n(8.99 x 109 N.
|x 107 N/C
m)
Transcribed Image Text:A uniformly charged insulating rod of length 19.5 cm is bent into the shape of a semicircle as shown in the figure below. The rod has a total charge of -7.50 µC.
(a) Find the magnitude of the electric field at O, the center of the semicircle.
(b) Find the direction of the electric field at o, the center of the semicircle.
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