To analyze the statements, let's break them down one by one with respect to the potential \( V = 0 \) at point P, defined against the potential at infinity being zero.
I. The force on a test charge at point P is zero.
The electric potential \( V \) can be zero at a point even if there are electric fields present. The electric field \( \vec{E} \) is related to the potential \( V \) by \( \vec{E} = -\nabla V \). If the potentials due to the charges combine to give \( V = 0 \) at point P, it does not imply that the electric field (and therefore the force) is zero. If the electric field at point P is not zero, then a test charge placed at P would feel a force. So, this statement is False.
II. The charges q1, q2, and q3 must be of the same sign but different magnitudes.
For the total electric potential at point P to be zero, it is not necessary for the charges to be of the same sign. The charges can be of different signs so long as their respective contributions to the potential cancel out at that specific point. Thus, it is possible to achieve a zero potential with charges of opposite signs. Therefore, this statement is False.
III. The net work needed to bring a test charge from infinity to P is zero.
The work done in bringing a test charge \( q \) from infinity (where \( V = 0 \)) to point P would be given by \( W = q \cdot (V_P - V_{\infty}) \). Since both potentials are zero at infinity and at point P (\( V_P = 0 \) and \( V_{\infty} = 0 \)), the work done would indeed be zero. Therefore, this statement is True.
Conclusion
- Statement I is False.
- Statement II is False.
- Statement III is True.
The only true statement is III alone.
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