Two spheres A and B of radius 4 cm and 6 cm are given charges of `80 muC and 40 muC`, respectively. If they are connected by a fine wire, then the amount of charge flowing from one to the other is

Options

A. `20 muC` from A to B
B. `20 muC` from B to A
C. `32 muC` from B to A
D. `32 muC` from A to B

Answer

To solve the problem, we need to determine how much charge flows from sphere A to sphere B when they are connected by a wire. We will follow these steps:

### Step 1: Understand the Initial Charges
- Sphere A has a charge \( Q_A = 80 \, \mu C \)
- Sphere B has a charge \( Q_B = 40 \, \mu C \)

### Step 2: Calculate the Total Charge
The total charge \( Q_{total} \) when the spheres are connected is the sum of the charges on both spheres:
\[ Q_{total} = Q_A + Q_B = 80 \, \mu C + 40 \, \mu C = 120 \, \mu C \]

### Step 3: Determine the Radii of the Spheres
- Radius of sphere A, \( r_A = 4 \, cm \)
- Radius of sphere B, \( r_B = 6 \, cm \)

### Step 4: Set Up the Relationship Between Charges and Radii
When the spheres are connected, they will reach the same electric potential. The potential \( V \) at the surface of a sphere is given by:
\[ V = \frac{k \cdot Q}{r} \]
where \( k \) is Coulomb's constant, \( Q \) is the charge, and \( r \) is the radius.

Since the potentials are equal after connecting the spheres:
\[ \frac{Q_A'}{r_A} = \frac{Q_B'}{r_B} \]
Let \( Q_A' \) and \( Q_B' \) be the final charges on spheres A and B, respectively.

### Step 5: Express the Final Charges in Terms of the Initial Charges
Using the conservation of charge, we know:
\[ Q_A' + Q_B' = Q_{total} = 120 \, \mu C \]
From the potential equality, we can express:
\[ \frac{Q_A'}{4} = \frac{Q_B'}{6} \]
Cross-multiplying gives:
\[ 6Q_A' = 4Q_B' \]
or
\[ \frac{Q_A'}{Q_B'} = \frac{4}{6} = \frac{2}{3} \]

### Step 6: Solve for Final Charges
Let \( Q_A' = 2x \) and \( Q_B' = 3x \). Then:
\[ 2x + 3x = 120 \, \mu C \]
\[ 5x = 120 \, \mu C \implies x = 24 \, \mu C \]
Thus, the final charges are:
\[ Q_A' = 2x = 48 \, \mu C \]
\[ Q_B' = 3x = 72 \, \mu C \]

### Step 7: Calculate the Charge Flow
The charge that flows from sphere A to sphere B is:
\[ \text{Charge Flow} = Q_A - Q_A' = 80 \, \mu C - 48 \, \mu C = 32 \, \mu C \]

### Final Answer
The amount of charge flowing from sphere A to sphere B is \( 32 \, \mu C \).

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