Consider a charge of +2.0 C placed at the origin of an X-Y co-ordinate system and a charge of -4.0 C placed 40.0 cm to the right. Where must a third charge be placed – between the charges, to the left of the origin, or beyond the second charge – to experience a net force of zero? Argue your case qualitatively without working out a solution. Consider both positive and negative charges.

the electric field of q1 E1

the electric field of q2 E2

Net electric field is vector sum of
electric fields due to q1 and q2.

the point to the left from origin:
E1←,→ E2 (can give zero)
the point between q1 and q2 :
E1 →,→ E2.(can’t give zero)
the point to the right from q2:
E1→,← E2. (E2>E1, =>can’t give zero).
Let's find the location of the point where electric field is zero.
k•q1•q3/r1² = k•q2•q3/r2²,
r1 =x,
r2 = x+d, where d = 0.4 m.
2/x² = 4/(x+d)²,
x² - 2•d•x - d² = 0.
Solve for x.
x1 = 0.954 m (to the left from origin - both for positive and negative q3)
x2 = - 0.164 m (is unsuitable)

To determine where a third charge must be placed to experience a net force of zero, we need to consider the electric forces exerted by the other charges. Here's a qualitative analysis:

1. Positive third charge: If the third charge is positive, it will experience attractive forces towards both the positive charge at the origin and the negative charge to the right. Since the positive charge at the origin is closer, the net force on the third charge will be towards the origin. Therefore, the third charge should be placed to the left of the origin to experience a net force of zero.

2. Negative third charge: If the third charge is negative, it will experience repulsive forces from both the positive charge at the origin and the negative charge to the right. Since the negative charge to the right is closer, the net force on the third charge will be away from the right charge. Therefore, the third charge should be placed beyond the second charge to experience a net force of zero.

In summary, a positive third charge should be placed to the left of the origin, while a negative third charge should be placed beyond the second charge to experience a net force of zero in this setup.