To determine the charges on the spheres [tex]\(X\)[/tex] and [tex]\(Y\)[/tex] after placing a metal rod in contact with both spheres, let's go through the problem step-by-step.
1. Initial Charges:
- Sphere [tex]\(X\)[/tex] is initially uncharged.
- Sphere [tex]\(Y\)[/tex] is initially positively charged.
2. Connecting with a Conductor:
- When the metal rod, which is a conductor, is placed in contact with both spheres, charge can flow freely between the two spheres.
3. Charge Distribution:
- Since the spheres are identical and the rod is a conductor, the charges will redistribute themselves evenly between the two spheres. This is because electric charges repel each other and try to move as far apart as possible, leading to an even distribution in identical spheres connected by a conductor.
4. Conservation of Charge:
- Total charge before contact must equal the total charge after contact. Let's suppose we represent the initial positive charge on sphere [tex]\(Y\)[/tex] as [tex]\(+1\)[/tex] unit for simplicity.
- The initial charge on sphere [tex]\(X\)[/tex] is [tex]\(0\)[/tex].
- Therefore, the total initial charge is:
[tex]\[
\text{Total charge} = 0 + 1 = 1 \text{ unit}
\][/tex]
5. Redistribution of Charge:
- Since the charge must be evenly distributed between the two spheres and the total charge is [tex]\(1\)[/tex] unit, each sphere will end up with half of the total charge.
- Hence, charge on each sphere after redistribution will be:
[tex]\[
\text{Charge on } X = \frac{1}{2} \text{ unit}
\][/tex]
[tex]\[
\text{Charge on } Y = \frac{1}{2} \text{ unit}
\][/tex]
6. Final Charges:
- The final charge on sphere [tex]\(X\)[/tex] is [tex]\(+0.5\)[/tex] units.
- The final charge on sphere [tex]\(Y\)[/tex] is [tex]\(+0.5\)[/tex] units.
- Therefore, both spheres [tex]\(X\)[/tex] and [tex]\(Y\)[/tex] end up with positive charges after the rod is placed in contact with them.
Based on the calculations, the correct answer is:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline & \text{charge on } X & \text{charge on } Y \\
\hline A & \text{positive} & \text{positive} \\
B & \text{positive} & \text{uncharged} \\
C & \text{uncharged} & \text{positive} \\
D & \text{uncharged} & \text{uncharged} \\
\hline
\end{tabular}
\][/tex]
The correct choice corresponds to option (A): positive and positive.
