25: A Test Charge Determines Charge on Insulating and Conducting Balls

INTRO: 
When a test charge is brought near a charged object, we know from Coulomb's law that it will experience a net force (either attractive or repulsive, depending on the nature of the object's charge). A test charge may also experience an electric force when brought near a neutral object. Any attraction of a neutral insulator or neutral conductor to a test charge must occur through induced polarization. In an insulator, the electrons are bound to their molecules. Though they cannot move freely throughout the insulator, they can shift slightly, creating a rather weak net attraction to a test charge that is brought close to the insulator's surface. In a conductor, free electrons will accumulate on the surface of the conductor nearest the positive test charge. This will create a strong attractive force if the test charge is placed very close to the conductor's surface.

Consider three plastic balls (A, B, and C), each carrying a uniformly distributed charge equal to either +Q, -Q or zero, and an uncharged copper ball (D). A positive test charge (T) experiences the forces shown in the figure when brought very near to the individual balls. The test charge T is strongly attracted to A, strongly repelled from B, weakly attracted to C, and strongly attracted to D.


Assume throughout this problem that the balls are brought very close together.

Part A: 
What is the nature of the force between balls A and B?

a) strongly attractive
b) strongly repulsive
c) weakly attractive
d) neither attractive nor repulsive

SOLUTION:
We begin by determining the net charges of balls A & B based on the reactions of the test charge near the balls, respectively 
We know that the test charge is positively charged
since there is a strongly attractive force between ball A and the test charge, the nature of the net charge of ball A must be negative
A = -Q
since there is a strongly repulsive force between ball B and the test charge, the nature of the net charge of ball B must be positive
B = +Q

so the nature of the force between a negative ball A and a positive ball B would be strongly attractive, or option a)

Part B:
What is the nature of the force between balls A and C?

a) strongly attractive
b) strongly repulsive
c) weakly attractive
d) neither attractive nor repulsive 

SOLUTION:
We already know the net charge of ball A from the previous part,
A = -Q
& we must also determine the net charge of ball C
In the intro, it is stated that "A test charge may also experience an electric force when brought near a neutral object... In an insulator, the electrons are bound to their molecules... they can shift slightly, creating a rather weak net attraction to a test charge that is brought close to the insulator's surface"
what this means is that C is a neutral insulator, based on its weak net attraction to the test charge. 
so, no matter the charge of the ball brought near to ball C, as long as it has any charge it will result in a weakly attractive force between the two balls, or option c)

Part C:
What is the nature of the force between balls A and D?

a) attractive
b) repulsive
c) neither attractive nor repulsive 

SOLUTION:
We already know the net charge of ball A from part A,
A = -Q
& we must use this to determine the nature of the forces between that ball and this uncharged copper ball D
In the intro it states that "In a conductor, free electrons will accumulate on the surface of the conductor nearest the positive test charge. This will create a strong attractive force if the test charge is placed very close to the conductor's surface."
so when ball A is brought close to ball D, the nature of the surface charge density on the side of ball D that is closest to ball A will be positive
This will result in an attractive force between the balls, or option a)

Part D:
What is the nature of the force between balls D and C?

a) attractive
b) repulsive
c) neither attractive nor repulsive 

SOLUTION:
C is a neutral insulator which will only react to charges brought near to it, as we determined in part c
and D is a neutral copper conductor which will only react to charges brought near to it as well
but since neither ball has a charge, when they are brought near to each other, it will result in neither an attractive nor a repulsive force, or option c)