Q.9
A cube of unit volume (1 m³) contains $35\times {10}^7$ photons of frequency ${10}^{15}\text{Hz}$. 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 $\alpha \times {10}^{-9}\text{T}$. Given:

• Permeability of free space ${\mu }_0=4\pi \times {10}^{-7}\text{Tm/A}$,

• Planck’s constant $h=6\times {10}^{-34}\text{Js}$,

• $\pi =\frac{22}{7}$.

The value of $\alpha$ is ______.

Answer Range: [21 to 25]

1. Understanding the Problem

We are given:

• A cube of volume $V=1{\text{m}}^3$ containing $N=35\times {10}^7$ photons.

• Each photon has frequency $\nu ={10}^{15}\text{Hz}$.

• The total energy of these photons is equivalent to the energy density of an electromagnetic (EM) wave in the same volume.

• We need to find the amplitude $B_0$ of the magnetic field in the EM wave, expressed as $\alpha \times {10}^{-9}\text{T}$.

2. Calculate Total Energy of Photons

The energy of one photon is:

$${\mathrm{<span\; style="background-color:\; white;">E</span>}}_{\text{<span style="background-color: white;">photon</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">h</span>}\mathrm{<span\; style="background-color:\; white;">\nu </span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">34</span>}}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">15</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">19</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">J</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

Total energy of all photons:

$${\mathrm{<span\; style="background-color:\; white;">E</span>}}_{\text{<span style="background-color: white;">total</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">N</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">E</span>}}_{\text{<span style="background-color: white;">photon</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">35</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">7</span>}}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">6</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">19</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">210</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">12</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">J</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

3. Relate Photon Energy to EM Wave Energy Density

The energy density $u$ of an EM wave is the sum of the electric and magnetic field energy densities:

$$\mathrm{<span\; style="background-color:\; white;">u</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}{\mathrm{<span\; style="background-color:\; white;">ϵ</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}{\mathrm{<span\; style="background-color:\; white;">E</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">+</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}{\mathrm{<span\; style="background-color:\; white;">\mu </span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}}{\mathrm{<span\; style="background-color:\; white;">B</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

For an EM wave, $E_0=c{B}_0$, where $c$ is the speed of light ($c\approx 3\times {10}^8\text{m/s}$). Substituting $E_0$:

$$\mathrm{<span\; style="background-color:\; white;">u</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}{\mathrm{<span\; style="background-color:\; white;">ϵ</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">c</span>}{\mathrm{<span\; style="background-color:\; white;">B</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">+</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}{\mathrm{<span\; style="background-color:\; white;">\mu </span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}}{\mathrm{<span\; style="background-color:\; white;">B</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">B</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{\mathrm{<span\; style="background-color:\; white;">2</span>}{\mathrm{<span\; style="background-color:\; white;">\mu </span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">+</span>{\mathrm{<span\; style="background-color:\; white;">\mu </span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}{\mathrm{<span\; style="background-color:\; white;">ϵ</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">.</span>}$$

Using $c^2=\frac{1}{{\mu }_0{ϵ}_0}$, this simplifies to:

$$\mathrm{<span\; style="background-color:\; white;">u</span>}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">B</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{{\mathrm{<span\; style="background-color:\; white;">\mu </span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

The total energy in volume $V=1{\text{m}}^3$ is:

$${\mathrm{<span\; style="background-color:\; white;">E</span>}}_{\text{<span style="background-color: white;">total</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">u</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">B</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{{\mathrm{<span\; style="background-color:\; white;">\mu </span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

Equate this to the total photon energy:

$$\frac{{\mathrm{<span\; style="background-color:\; white;">B</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{{\mathrm{<span\; style="background-color:\; white;">\mu </span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">210</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">12</span>}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

4. Solve for $B_0$

Given ${\mu }_0=4\pi \times {10}^{-7}\text{Tm/A}$ and $\pi =\frac{22}{7}$:

$${\mathrm{<span\; style="background-color:\; white;">B</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">210</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">12</span>}}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">4</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">7</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">840</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">19</span>}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

Substitute $\pi =\frac{22}{7}$:

$${\mathrm{<span\; style="background-color:\; white;">B</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">840</span>}<span\; style="background-color:\; white;">\times </span>\frac{\mathrm{<span\; style="background-color:\; white;">22</span>}}{\mathrm{<span\; style="background-color:\; white;">7</span>}}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">19</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">2640</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">19</span>}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

Take the square root:

$${\mathrm{<span\; style="background-color:\; white;">B</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}<span\; style="background-color:\; white;">=</span>\sqrt{\mathrm{<span\; style="background-color:\; white;">2640</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">19</span>}}}<span\; style="background-color:\; white;">=</span>\sqrt{\mathrm{<span\; style="background-color:\; white;">26.4</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">18</span>}}}<span\; style="background-color:\; white;">=</span>\sqrt{\mathrm{<span\; style="background-color:\; white;">26.4</span>}}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">9</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">T</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

Approximate $\sqrt{26.4}\approx 5.14$:

$${\mathrm{<span\; style="background-color:\; white;">B</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}<span\; style="background-color:\; white;">\approx </span>\mathrm{<span\; style="background-color:\; white;">5.14</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">9</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">T</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

Thus, $\alpha \approx 5.14$.

5. Reconcile with Expected Answer

The expected answer range is [21 to 25], but our calculation gives $\alpha \approx 5.14$. This discrepancy suggests:

• The problem might consider the peak energy density rather than the average. For a sinusoidal EM wave, the peak energy density is twice the average:

  $${\mathrm{<span\; style="background-color:\; white;">u</span>}}_{\text{<span style="background-color: white;">peak</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">2</span>}{\mathrm{<span\; style="background-color:\; white;">u</span>}}_{\text{<span style="background-color: white;">avg</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">210</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">12</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">420</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">12</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">J/m</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

  Recalculating:

  $$\frac{{\mathrm{<span\; style="background-color:\; white;">B</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{{\mathrm{<span\; style="background-color:\; white;">\mu </span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">420</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">12</span>}}\text{<span style="background-color: white;">\hspace{0.25em}\hspace{0.05em}</span>}<span\; style="background-color:\; white;">⟹</span>\text{<span style="background-color: white;">\hspace{0.25em}\hspace{0.05em}</span>}{\mathrm{<span\; style="background-color:\; white;">B</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1680</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">19</span>}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$$${\mathrm{<span\; style="background-color:\; white;">B</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}<span\; style="background-color:\; white;">=</span>\sqrt{\mathrm{<span\; style="background-color:\; white;">1680</span>}<span\; style="background-color:\; white;">\times </span>\frac{\mathrm{<span\; style="background-color:\; white;">22</span>}}{\mathrm{<span\; style="background-color:\; white;">7</span>}}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">19</span>}}}<span\; style="background-color:\; white;">=</span>\sqrt{\mathrm{<span\; style="background-color:\; white;">5280</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">19</span>}}}<span\; style="background-color:\; white;">\approx </span>\mathrm{<span\; style="background-color:\; white;">7.27</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">9</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">T</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

  This gives $\alpha \approx 7.27$, still outside the expected range.

• Alternatively, the problem might involve standing waves or other configurations where the energy density is higher. However, without additional information, the most accurate calculation based on the given data yields $\alpha \approx 5.14$.

6. Final Answer

Based on standard EM wave theory and the given data, the value of $\alpha$ is:

\boxed{5.14}

(Note: The expected range [21 to 25] may imply additional context or corrections not provided in the problem statement.)