A magnetron in a microwave oven emits electromagnetic wave with frequency $f=2450$ MHz. The magnitude of the magnetic field required for electron to move in circular path with this frequency is:

Question

Accepted Answer

To determine the magnitude of the magnetic field required for an electron to move in a circular path with a given frequency in a magnetron, we can use the relationship between the cyclotron frequency and the magnetic field. The cyclotron frequency $ f_c $ is given by the formula:

$$ f_c = \frac{eB}{2\pi m} $$

where:
- $ e $ is the elementary charge ($1.602 \times 10^{-19}$ C),
- $ B $ is the magnetic field strength,
- $ m $ is the mass of the electron ($9.109 \times 10^{-31}$ kg),
- $ f_c $ is the frequency of the electron's circular motion.

Given that the frequency $ f = 2450 $ MHz, we need to convert this to Hz:

$$ f = 2450 \times 10^6 \, \text{Hz} $$

Setting $ f_c = f $, we have:

$$ 2450 \times 10^6 = \frac{(1.602 \times 10^{-19})B}{2\pi (9.109 \times 10^{-31})} $$

Solving for $ B $:

$$ B = \frac{2450 \times 10^6 \times 2\pi \times 9.109 \times 10^{-31}}{1.602 \times 10^{-19}} $$

$$ B \approx 0.0877 \, \text{T} $$

Thus, the magnitude of the magnetic field required is approximately 0.0877 T, which corresponds to Option 3.

**Explanation for Other Options:**

1. **Option 1 (0.0438 T):** This value is approximately half of the correct magnetic field strength. It does not satisfy the equation derived from the cyclotron frequency formula when substituting the given frequency of 2450 MHz.

2. **Option 2 (0.0270 T):** This value is significantly lower than the correct magnetic field strength. It would result in a cyclotron frequency much lower than 2450 MHz, which does not match the given conditions.

3. **Option 4 (0.05415 T):** This value is also lower than the correct magnetic field strength. It would not provide the necessary cyclotron frequency of 2450 MHz for the electron's circular motion.

In conclusion, Option 3 is the only value that satisfies the condition for the electron to move in a circular path with the specified frequency of 2450 MHz, making it the correct answer.