Question

Consider the electromagnetic field produced at a given distance from a light source. Which of the following statements are true? 

Check all that apply. 

 1. The intensity of the wave is inversely proportional to the square of the fields. 

2. The intensity of the wave is proportional to the square of the fields. 

3. The intensity of the wave is proportional to the speed of light. 

4. The intensity of the wave is inversely proportional to the speed of light.

Answer 1

Concepts and reason

The concepts used to solve this problem are intensity of electromagnetic wave.

Initially, find the expression for the intensity of an electromagnetic wave.

Then, find the wrong options by using the intensity expression.

Finally, find the correct option using intensity expression.

Fundamentals

Electromagnetic wave consists of both electric and magnetic field.

A light source will radiate energy in all directions.

The intensity decreases in proportion to distance from the object squared.

Due to conservation law,

The net power is ,

$p = \int I \cdot dA$

Here, $I$ is the intensity, $P$ is the power, and $dA$ is the differential element of closed surface.

Integrate over surface of uniform intensity,$I$ the eqution becomes,

$\begin{array}{c}\\P = \left| I \right| \cdot {A_{{\rm{surf}}}}\\\\ = \left| I \right| \cdot 4\pi {r^2}\\\end{array}$

Here, the surface area of sphere is $4\pi {r^2}$ because the source will radiate in all directions.

Rearrange the above equation,

$\left| I \right| = \frac{P}{{4\pi {r^2}}}$

Here, $\left| I \right|$ is the modulus of intensity, and $r$ is the radius.

If electric field is complex amplitude of the electric field,then the time averaged energy density of the wave is,

$\left\langle U \right\rangle = \frac{{n{\varepsilon _0}}}{2}{\left| E \right|^2}$

Here, $\left\langle U \right\rangle$ is the total energy, $n$ is the refractive index, $\left| E \right|$ is the complex electric field, and ${\varepsilon _0}$ is the vacuum permittivity.

The local intensity is obtined by multiplying the wave velocity.

The expression for wave velocity is,

$v = \frac{c}{n}$

Here, $v$ is the velocity and $c$ is the speed of light.

The rearranged expression for intensity is,

$I = \frac{{cn{\varepsilon _0}}}{2}{\left| E \right|^2}$

Here, $I$ is the intensity.

The epression for itensity is,

$I = \frac{{cn{\varepsilon _0}}}{2}{\left| E \right|^2}$

The wrong options are,

• Intensity of wave is inversely proportionl to the square of the fields.

• The intensity of wave is inversely proportional to the speed of light.

From the above expresson intensity is directly proportionl to the square of the fields. And also the speed of light is directly proportionl to the intensity.

Thus, the inversely proportionl option is wrong.

The correct option is,

• Intensity of wave is proportionl to the square of the fields.

The correct options is,

• The intensity of wave is proportional to the speed of light.

From the expression,

$I = \frac{{cn{\varepsilon _0}}}{2}{\left| E \right|^2}$

Clearly shows that the intensity is directly proportionl to the speed of light.

Ans:

The correct option is Intensity of wave is proportionl to the square of the fields.

The correct option is intensity is directly proportionl to the speed of light.