A neutral sphere of radius r and density `rho` is placed in a uniform electric field E that exists on the earth's surface in the vertically upward direction. If atomic number and the mass number of the material of the sphere are Z and A respectively, then the fraction of electrons that should be removed from the sphere for it to remain in equilibrium is [Assume that the sphere is able to hold the necessary charge without any leakage. Here `N_(A)` - Avogadro number]

Options

A. `n/(n_("total")) = (rho g A)/(eEN_A Z)`
B. `n/(n_("total")) = (4 g A)/(pieEN_A Z)`
C. `n/(n_("total")) = ( g A)/(eEN_A Z)`
D. `n/(n_("total")) = (pi rho g A)/(3eEN_A Z)`

Answer

To solve the problem, we need to find the fraction of electrons that should be removed from a neutral sphere placed in a uniform electric field in order for it to remain in equilibrium. Let's break down the solution step by step.

### Step 1: Understand the forces acting on the sphere
The sphere experiences two main forces:
1. **Gravitational Force (downward)**: This force can be expressed as:
\[ F_g = mg \]
where \( m \) is the mass of the sphere and \( g \) is the acceleration due to gravity.

2. **Electric Force (upward)**: This force is due to the electric field and can be expressed as:
\[ F_e = QE \]
where \( Q \) is the charge on the sphere and \( E \) is the electric field strength.

### Step 2: Calculate the mass of the sphere
The mass \( m \) of the sphere can be calculated using its volume and density:
\[ m = \text{Volume} \times \text{Density} = \left(\frac{4}{3} \pi r^3\right) \rho \]

### Step 3: Determine the charge required for equilibrium
For the sphere to remain in equilibrium, the upward electric force must balance the downward gravitational force:
\[ mg = QE \]
Substituting \( Q \) with \( n \cdot e \) (where \( n \) is the number of electrons removed and \( e \) is the charge of an electron):
\[ mg = n \cdot e \cdot E \]

### Step 4: Solve for the number of electrons removed
From the equilibrium condition, we can express \( n \):
\[ n = \frac{mg}{eE} \]

### Step 5: Calculate the total number of electrons in the sphere
The total number of electrons in the sphere can be calculated using the number of moles:
\[ \text{Number of moles} = \frac{m}{A} \]
where \( A \) is the mass number of the material. The total number of electrons is then:
\[ \text{Total electrons} = \text{Number of moles} \times N_A \times Z = \frac{m}{A} \times N_A \times Z \]

### Step 6: Substitute mass into the total number of electrons
Substituting \( m \):
\[ \text{Total electrons} = \frac{\left(\frac{4}{3} \pi r^3 \rho\right)}{A} \times N_A \times Z \]

### Step 7: Calculate the fraction of electrons removed
The fraction of electrons removed is given by:
\[ \text{Fraction} = \frac{n}{\text{Total electrons}} = \frac{\frac{mg}{eE}}{\frac{4}{3} \pi r^3 \rho}{A} \times N_A \times Z \]
Substituting \( m \):
\[ \text{Fraction} = \frac{\frac{\left(\frac{4}{3} \pi r^3 \rho\right) g}{eE}}{\frac{4}{3} \pi r^3 \rho}{A} \times N_A \times Z \]
After simplification, we find:
\[ \text{Fraction} = \frac{gA}{eEN_AZ} \]

### Final Result
Thus, the fraction of electrons that should be removed from the sphere for it to remain in equilibrium is:
\[ \text{Fraction} = \frac{gA}{eEN_AZ} \]