Four identical charges are placed at the points `(1,0,0), (0,1,0),(-1,0,0)`, and `(0,-1,0)`. Then,

Options

A. the potential at the origin is zero
B. the electric field at the origin is not zero.
C. the potential at all points on the z - axis, other than the origin, is zero
D. the field at all points on the z - axis, other than the origin acts along the z - axis.

Explanation

To solve the question regarding the four identical charges placed at the points (1,0,0), (0,1,0), (-1,0,0), and (0,-1,0), we need to analyze the electric potential and electric field at the origin (0,0,0) and along the z-axis.

### Step-by-Step Solution:

1. **Identify the Charges and Their Positions**:
- We have four identical charges (let's denote them as Q) located at:
- Charge 1 at (1,0,0)
- Charge 2 at (0,1,0)
- Charge 3 at (-1,0,0)
- Charge 4 at (0,-1,0)

2. **Calculate the Electric Potential at the Origin**:
- The electric potential \( V \) at a point due to a point charge is given by the formula:
\[ V = k \frac{Q}{r} \]
where \( k \) is Coulomb's constant and \( r \) is the distance from the charge to the point of interest.
- The distance from each charge to the origin (0,0,0) is 1 unit (since the charges are at a distance of 1 from the origin).
- Therefore, the potential at the origin due to each charge is:
\[ V_{total} = k \frac{Q}{1} + k \frac{Q}{1} + k \frac{Q}{1} + k \frac{Q}{1} = 4kQ \]

3. **Determine the Electric Field at the Origin**:
- The electric field \( E \) due to a point charge is given by:
\[ E = k \frac{Q}{r^2} \]
- The direction of the electric field due to each charge will be directed away from the charge if the charge is positive.
- For the charges at (1,0,0) and (-1,0,0), the electric fields at the origin will cancel each other out:
\[ E_x = k \frac{Q}{1^2} - k \frac{Q}{1^2} = 0 \]
- Similarly, for the charges at (0,1,0) and (0,-1,0):
\[ E_y = k \frac{Q}{1^2} - k \frac{Q}{1^2} = 0 \]
- Therefore, the total electric field at the origin is:
\[ E_{total} = E_x + E_y = 0 + 0 = 0 \]

4. **Electric Potential at Points on the Z-axis**:
- The potential at any point on the z-axis (other than the origin) can be calculated similarly, and since the distances from the charges to any point on the z-axis will still be greater than 1, the potential will still be positive but will not equal zero.

5. **Electric Field Along the Z-axis**:
- The electric field at any point along the z-axis will not be zero. The contributions from the charges will not cancel out, and the field will point along the z-axis.

### Conclusion:
- The potential at the origin is \( 4kQ \).
- The electric field at the origin is \( 0 \).
- The potential at all points on the z-axis other than the origin is \( 0 \).
- The electric field along the z-axis is not zero.

### Final Answers:
- The potential at the origin is \( 4kQ \).
- The electric field at the origin is \( 0 \).
- The potential at all points on the z-axis other than the origin is \( 0 \).
- The electric field at all points on the z-axis other than the origin is directed along the z-axis.