How much positive charge should be given to the earth so as to have the same potential as that of a positively charged sphere of `1 muC` and radius 1 cm . ( Radius of earth = 6400km )

Options

A. 600 C
B. 640 C
C. 340 C
D. 240 C

Answer

To solve the problem of how much positive charge should be given to the Earth so that it has the same potential as a positively charged sphere of 1 µC and radius 1 cm, we can follow these steps:

### Step 1: Understand the formula for electric potential
The electric potential \( V \) due to a point charge is given by the formula:
\[ V = \frac{kQ}{r} \]
where:
- \( V \) is the electric potential,
- \( k \) is Coulomb's constant (\( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)),
- \( Q \) is the charge,
- \( r \) is the distance from the charge.

### Step 2: Set up the equation for the sphere and the Earth
Let:
- \( Q_1 \) be the charge on the Earth,
- \( R_E \) be the radius of the Earth (\( 6400 \, \text{km} = 6.4 \times 10^6 \, \text{m} \)),
- \( Q_2 \) be the charge of the sphere (\( 1 \, \mu\text{C} = 1 \times 10^{-6} \, \text{C} \)),
- \( r \) be the radius of the sphere (\( 1 \, \text{cm} = 0.01 \, \text{m} \)).

The potential of the Earth \( V_E \) and the potential of the sphere \( V_S \) can be expressed as:
\[ V_E = \frac{kQ_1}{R_E} \]
\[ V_S = \frac{kQ_2}{r} \]

### Step 3: Equate the potentials
Since we want the potential of the Earth to be equal to the potential of the sphere:
\[ \frac{kQ_1}{R_E} = \frac{kQ_2}{r} \]

### Step 4: Simplify the equation
We can cancel \( k \) from both sides:
\[ \frac{Q_1}{R_E} = \frac{Q_2}{r} \]

### Step 5: Solve for \( Q_1 \)
Rearranging the equation gives:
\[ Q_1 = Q_2 \cdot \frac{R_E}{r} \]

### Step 6: Substitute the values
Substituting \( Q_2 = 1 \times 10^{-6} \, \text{C} \), \( R_E = 6.4 \times 10^6 \, \text{m} \), and \( r = 0.01 \, \text{m} \):
\[ Q_1 = (1 \times 10^{-6}) \cdot \frac{6.4 \times 10^6}{0.01} \]
\[ Q_1 = (1 \times 10^{-6}) \cdot (6.4 \times 10^8) \]
\[ Q_1 = 6.4 \times 10^2 \, \text{C} \]
\[ Q_1 = 640 \, \text{C} \]

### Conclusion
The amount of positive charge that should be given to the Earth to have the same potential as that of the positively charged sphere is **640 C**.

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