Two identical coils carry equal currents and have a common center, but their planes are at right angles to each other. What is the magnitude of the resultant magnetic field at the center, if field due to one coil alone is B ?

Question

Accepted Answer

To determine the magnitude of the resultant magnetic field at the center of two identical coils carrying equal currents and having a common center, with their planes at right angles to each other, we need to consider the vector nature of magnetic fields.

1. **Explanation for Option 4 ($\sqrt{2}B$):**

When two magnetic fields are perpendicular to each other, the resultant magnetic field can be found using the Pythagorean theorem. If the magnetic field due to one coil alone is $ B $, and since the coils are at right angles, the fields are perpendicular. Therefore, the resultant magnetic field $ B_{\text{resultant}} $ is given by:

$$
   B_{\text{resultant}} = \sqrt{B^2 + B^2} = \sqrt{2B^2} = \sqrt{2}B
   $$

This calculation shows that the magnitude of the resultant magnetic field at the center is indeed $ \sqrt{2}B $, which corresponds to Option 4.

2. **Explanation for why other options are incorrect:**

- **Option 1 (B/2):** This option suggests that the resultant magnetic field is half of the field due to one coil. This is incorrect because the fields are perpendicular and should be combined using the Pythagorean theorem, not simply averaged or halved.

- **Option 2 (2B):** This option implies that the fields add linearly as if they were in the same direction. However, since the fields are perpendicular, they do not add directly. The correct method is to use vector addition, which results in $ \sqrt{2}B $, not $ 2B $.

- **Option 3 ($B/\sqrt{2}$):** This option suggests a reduction in the field strength, which is incorrect. The resultant field from two perpendicular fields of equal magnitude is greater than the field due to one coil alone, not less. The correct resultant is $ \sqrt{2}B $, not a reduced value like $ B/\sqrt{2} $.

In conclusion, the correct answer is Option 4, $ \sqrt{2}B $, because it accurately represents the resultant magnetic field when two perpendicular magnetic fields of equal magnitude are combined.