A non-conducting sphere of uniform charge density = +5 µC/m3 and outer radius R

Question

A non-conducting sphere of uniform charge density = +5 µC/m3 and outer radius R = 12 m has a smaller spherical cavity of radius a = 4 m cut out of its interior. This cavity is not centered-- the center of the larger sphere and the center of the cavity are a distance b = 4.8 m apart, as shown below.

Find the electric field inside the cavity at the following locations: 
(Hint: While this object as a whole does not have radial symmetry-- think superposition.)

A non-conducting sphere of uniform charge density ? = +5 muC/m^3 and outer radius R = 12 m has a smaller spherical cavity of radius a = 4 m cut out of its interior. This cavity is not centered-- the center of the larger sphere and the center of the cavity are a distance b = 4.8 m apart, as shown below. Find the electric field inside the cavity at the following locations: (Hint: While this object as a whole does not have radial symmetry-- think superposition.) a.) A distance of a/2 to the left of the cavity's center. Vector E = ________ N/C Hat i + ________ N/C Hat j + ________ N/C Hat k b.) A distance of a/2 above the cavity's center. Vector E = ________ N/C Hat i + ________ N/C Hat j + ________ N/C Hat k Make sure to compare/contrast your answers. (Challenge: Can you find a general expression for the electric field within the cavity?) c.) Repeat part a, if this object were instead a conductor which has the same total charge. Vector E = ________ N/C Hat i + ________ N/C Hat j + ________ N/C Hat k

Explanation / Answer

The "superposition" idea is that the sphere of positive charge with a small cavity is the same as a full sphere of positive charge (i.e. with no cavity) superimposed with a small sphere (i.e. same size as the cavity) of negative charge (with same charge density). For the area of superposition, the total charge is zero (where the positive and negative superimpose).

This makes calculating the E field much easier, because the magnitude of each E field (one from the full positive sphere, the other from the small negative sphere) is simply   k q / r^2   where k is a constant, q is the total charge enclosed by the given radius and r is the distance from the center of the sphere to the given point.

(I misread the question and calculated parts A and B wrong).

Answer to part C:     0         0           0               E field inside a conductor is zero.

 

 

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