A vertical straight conductor carries a current vertically upwards. A point `P` lies to the east of it at a small distance and another point `Q` lies to the west at the same distance. The magnetic field at `P` is

Options

A. Greater than at `Q`
B. Same as at `Q`
C. Less than at `Q`
D. Greater or less than at `Q` depending upon the magnetic field of the current.

Answer

To solve the problem, we need to determine the magnetic field at points P and Q due to a vertical straight conductor carrying a current upwards.

### Step-by-Step Solution:

1. **Understand the Configuration**:
- We have a vertical straight conductor carrying a current upwards.
- Point P is located to the east of the conductor, and point Q is to the west of the conductor, both at the same distance from the wire.

2. **Use the Right-Hand Rule**:
- To determine the direction of the magnetic field around the conductor, we apply the right-hand rule.
- If you point your thumb upwards (in the direction of the current), your fingers will curl around the wire in the direction of the magnetic field lines.

3. **Determine the Magnetic Field Direction**:
- At point P (east of the wire), the magnetic field will be directed into the screen (or page).
- At point Q (west of the wire), the magnetic field will be directed out of the screen (or page).

4. **Calculate the Magnitude of the Magnetic Field**:
- The magnetic field (B) due to a long straight conductor at a distance (r) from it is given by the formula:
\[ B = \frac{\mu_0 I}{2\pi r} \]
- Here, \( \mu_0 \) is the permeability of free space, and \( I \) is the current. Since both points P and Q are at the same distance from the wire, the magnitudes of the magnetic fields at both points will be equal.

5. **Combine the Magnetic Fields**:
- At point P, the magnetic field due to the wire is directed into the screen, and the magnetic field due to the Earth (assumed to be directed downwards) will also be in the same direction (into the screen).
- At point Q, the magnetic field due to the wire is directed out of the screen, while the magnetic field due to the Earth is still directed downwards (which is opposite to the direction of the wire's field).
- Therefore, at point P, the two magnetic fields add together, while at point Q, they subtract.

6. **Conclusion**:
- Since the magnetic fields at point P add up, while at point Q they subtract, the magnetic field at point P will be greater than that at point Q.
- Thus, we conclude:
\[ B_P > B_Q \]

### Final Answer:
The magnetic field at point P is greater than the magnetic field at point Q.

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