A cube of unit volume contains $$35 \times 10^7$$ photons of frequency $$10^{15}$$ Hz. If the energy of all the photons is viewed as the average energy being contained in the electromagnetic waves within the same volume, then the amplitude of the magnetic field is $$\alpha \times 10^{-9}$$ T. Taking permeability of free space $$\mu_0 = 4\pi \times 10^{-7}$$ Tm/A, Planck's constant $$h = 6 \times 10^{-34}$$ Js and $$\pi = \frac{22}{7}$$, the value of $$\alpha$$ is ________.

The cube has volume $$V = 1 \;{\text{m}}^{3}$$, frequency of each photon $$\nu = 10^{15}\,{\text{Hz}}$$ and number of photons $$N = 35 \times 10^{7}$$.

Energy of one photon (Planck’s relation)
$$E_{\text{photon}} = h \nu = 6 \times 10^{-34}\,{\text{J\,s}}\; \times 10^{15}\,{\text{Hz}} = 6 \times 10^{-19}\,{\text{J}}.$$

Total energy contained in the cube
$$U = N \, E_{\text{photon}} = 35 \times 10^{7} \times 6 \times 10^{-19} = (35 \times 6) \times 10^{7-19} = 210 \times 10^{-12} = 2.10 \times 10^{-10}\,{\text{J}}.$$

Because the volume is $$1\,{\text{m}}^{3}$$, the average electromagnetic energy density is
$$u = \frac{U}{V} = 2.10 \times 10^{-10}\,{\text{J\,m}}^{-3}.$$

For a plane electromagnetic wave, the time-averaged total energy density is
$$u = \frac{B_0^{2}}{2\mu_0},$$
where $$B_0$$ is the amplitude of the magnetic field.

Solving for $$B_0$$:
$$B_0^{2} = 2\mu_0 u$$
$$B_0 = \sqrt{2\mu_0 u}.$$ Insert the given value $$\mu_0 = 4\pi \times 10^{-7}\,{\text{T\,m\,A}}^{-1} = \frac{88}{7}\times 10^{-7}\,{\text{T\,m\,A}}^{-1} \approx 1.257 \times 10^{-6}\,{\text{T\,m\,A}}^{-1}$$.

Compute the product:
$$2\mu_0 u = 2 \times 1.257 \times 10^{-6} \times 2.10 \times 10^{-10} = 5.28 \times 10^{-16}\,{\text{T}}^{2}.$$

Hence
$$B_0 = \sqrt{5.28 \times 10^{-16}} = \sqrt{5.28}\times 10^{-8}\,{\text{T}} \approx 2.30 \times 10^{-8}\,{\text{T}}.$$

Writing the amplitude in the required form $$B_0 = \alpha \times 10^{-9}\,{\text{T}},$$
$$\alpha = \frac{2.30 \times 10^{-8}}{10^{-9}} \approx 23.$$

Therefore, the value of $$\alpha$$ lies in the range $$21 \text{ to } 25$$, as specified.