Electric Field Potential Magnetic Field and Moment of Rotating Charges

To find the correct match for each entry in Column I, we analyze each configuration given in Column II:

Analysis of Configurations in Column II:

1. Configuration (p):
   • Charges and Geometry: Six charges arranged at the vertices of a regular hexagon. Three charges are $+q$ forming an equilateral triangle, and the other three are $-q$ forming another inverted equilateral triangle.
   • Electric Field ($E$): The electric fields due to the three symmetric $+q$ charges at the center $\text{M}$ cancel out to zero ($\vec{E}_+ = 0$). Similarly, the electric fields due to the three $-q$ charges cancel out ($\vec{E}_- = 0$). Thus, the net electric field is:
     $E = 0$
   • Electric Potential ($V$): All charges are at an equal distance $R$ from the center $\text{M}$. The sum of charges is $\sum q_i = 3(+q) + 3(-q) = 0$. Therefore:
     $V = \frac{1}{4\pi\varepsilon_0 R} \sum q_i = 0$
   • Magnetic Field ($B$) and Magnetic Moment ($\mu$): The line $\text{PQ}$ is perpendicular to the plane of the hexagon and passes through $\text{M}$. When rotated about $\text{PQ}$, all charges move in circles of radius $R$. The equivalent circular current due to the positive charges is $I_+ = 3\left(\frac{q\omega}{2\pi}\right)$, and due to the negative charges is $I_- = -3\left(\frac{q\omega}{2\pi}\right)$. The net circulating current is zero:
     $I_{\text{net}} = I_+ + I_- = 0 \implies B = 0 \quad \text{and} \quad \mu = 0$
   • Matches: (p) satisfies $E = 0$ and $B = 0$.

1. Configuration (q):
   • Charges and Geometry: Six charges are arranged symmetrically about $\text{M}$ along a line perpendicular to $\text{PQ}$. At each symmetric distance $r_i$ from $\text{M}$, there is a pair of opposite charges $+q$ and $-q$.
   • Electric Field ($E$): Since opposite charges lie on opposite sides of $\text{M}$, their electric fields at the midpoint $\text{M}$ point in the same direction and reinforce each other:
     $E \neq 0$
   • Electric Potential ($V$): At each distance $r_i$, the pair contributes $+q$ and $-q$, so their potentials cancel pairwise:
     $V = \sum \frac{k q_i}{r_i} = 0$
   • Magnetic Field ($B$) and Magnetic Moment ($\mu$): During rotation about $\text{PQ}$, each pair of charges at distance $r_i$ traces out the same circle of radius $r_i$. The net charge on any circle is $(+q) + (-q) = 0$, producing zero net circular current at each radius:
     $B = 0 \quad \text{and} \quad \mu = 0$
   • Matches: (q) satisfies $B = 0$.

1. Configuration (r):
   • Charges and Geometry: The charges are arranged symmetrically on two concentric coplanar rings with their common center at $\text{M}$. The net charge of the system is non-zero ($\sum q_i \neq 0$).
   • Electric Field ($E$): By circular symmetry of the equal-interval charge distribution on each ring:
     $E = 0$
   • Electric Potential ($V$): Since the net charge is non-zero:
     $V \neq 0$
   • Magnetic Field ($B$) and Magnetic Moment ($\mu$): The rotation axis $\text{PQ}$ passes perpendicularly through the common center $\text{M}$. The rotation creates concentric circular currents with a non-zero net circulating charge:
     $B \neq 0 \quad \text{and} \quad \mu \neq 0$
   • Matches: (r) satisfies $E = 0$, $V \neq 0$, and $\mu \neq 0$.

1. Configuration (s):
   • Charges and Geometry: Six charges are placed at the vertices and midpoints of the longer sides of a rectangle. The charge distribution is symmetric such that identical charges face each other across the center $\text{M}$, and the net charge is non-zero.
   • Electric Field ($E$): Equal charges are located diametrically opposite to each other across the center $\text{M}$, so their electric field vectors cancel pairwise:
     $E = 0$
   • Electric Potential ($V$): Since the total charge is non-zero:
     $V \neq 0$
   • Magnetic Field ($B$) and Magnetic Moment ($\mu$): Rotation about the line of symmetry $\text{PQ}$ produces non-zero equivalent current loops that generate a net magnetic field at $\text{M}$ and a net magnetic dipole moment:
     $B \neq 0 \quad \text{and} \quad \mu \neq 0$
   • Matches: (s) satisfies $E = 0$, $V \neq 0$, and $\mu \neq 0$.

1. Configuration (t):
   • Charges and Geometry: Two identical rings carrying charges such that the net charge on the left ring is $+3q$ and on the right ring is $-3q$. $\text{M}$ is the midpoint between the centers of the rings.
   • Electric Field ($E$): The fields from the positively charged ring and negatively charged ring point in the same direction at $\text{M}$:
     $E \neq 0$
   • Electric Potential ($V$): By anti-symmetry about the perpendicular bisector line $\text{PQ}$, the potential from the positive ring and negative ring cancel at every point on the plane of symmetry containing $\text{PQ}$:
     $V = 0$
   • Magnetic Field ($B$) and Magnetic Moment ($\mu$): Due to the anti-symmetric arrangement of charges about the rotation axis $\text{PQ}$, the magnetic field generated at $\text{M}$ and the total magnetic dipole moment cancel out:
     $B = 0 \quad \text{and} \quad \mu = 0$
   • Matches: (t) satisfies $B = 0$.

Matching Summary:

• (A) $E = 0$ $\to$ p, r, s
• (B) $V \neq 0$ $\to$ r, s
• (C) $B = 0$ $\to$ p, q, t
• (D) $\mu \neq 0$ $\to$ r, s