Two point charges Q and -3Q are placed at some distance apart. If the electric field at the location of Q is `vec (E )`, the field at the location of - 3Q is

Options

A. `vec(E )`
B. `-vec(E )`
C. `+ (vec( E))/(3)`
D. `-(vec(E ))/(3)`

Answer

To solve the problem, we need to find the electric field at the location of the charge -3Q, given that the electric field at the location of charge Q is \(\vec{E}\).

### Step-by-Step Solution:

1. **Understanding Electric Field due to Point Charges**:
The electric field \( \vec{E} \) due to a point charge \( Q \) at a distance \( r \) is given by the formula:
\[ \vec{E} = k \frac{Q}{r^2} \]
where \( k \) is Coulomb's constant.

2. **Identify the Charges and Their Positions**:
Let’s denote the position of charge \( Q \) as point A and the position of charge -3Q as point B. The distance between the two charges is \( r \).

3. **Calculate Electric Field at Point A (due to -3Q)**:
The electric field at point A (where charge Q is located) due to charge -3Q is directed towards charge -3Q (since it is negative). The magnitude of this electric field is:
\[ \vec{E}_{A} = k \frac{-3Q}{r^2} \]

4. **Given Electric Field at Point A**:
We are given that the electric field at the location of charge Q (point A) is \( \vec{E} \). This means:
\[ \vec{E} = k \frac{-3Q}{r^2} \]

5. **Calculate Electric Field at Point B (due to Q)**:
Now, we need to find the electric field at point B (where charge -3Q is located) due to charge Q. The electric field at point B due to charge Q is directed away from charge Q (since it is positive). The magnitude of this electric field is:
\[ \vec{E}_{B} = k \frac{Q}{r^2} \]

6. **Relate the Electric Fields**:
Since we have established that \( \vec{E} = k \frac{-3Q}{r^2} \), we can express \( Q \) in terms of \( \vec{E} \):
\[ Q = -\frac{E r^2}{3k} \]
Now substituting this into the expression for \( \vec{E}_{B} \):
\[ \vec{E}_{B} = k \frac{Q}{r^2} = k \frac{-\frac{E r^2}{3k}}{r^2} = -\frac{E}{3} \]

7. **Final Result**:
Therefore, the electric field at the location of charge -3Q is:
\[ \vec{E}_{B} = -\frac{\vec{E}}{3} \]

### Conclusion:
The electric field at the location of charge -3Q is \(-\frac{\vec{E}}{3}\).