Intensity Formula 

In wave optics, the intensity of light measures the energy transmitted from the source. The greater the energy emitted by the light source, the higher the intensity of the light. Intensity is a scalar quantity and is a crucial property of a light source. Learn more about intensity formula and its application in this article.

What is Intensity?

Intensity is the amount of energy a wave transmits per unit time across a unit area. It is also equivalent to the product of energy density and wave speed.Generally, intensity is measured in units of watts per square meter. The magnitude of intensity depends on the strength and amplitude of the propagating wave. Intensity is represented by the symbol I.

Intensity Formula 

The intensity formula describes how power (P) of a wave is distributed over a given area (A).The intensity of light is defined as the amount of power transferred per unit area. It is also referred to as the flux of radiant energy. The area is measured on a plane that is perpendicular to the direction in which the energy is propagating. It is mathematically represented as

I= P/A

​Where:

• I denotes intensity,
• P is the power or energy transmitted per unit time,
• A is the area over which the power is spread.

Intensity, measured in watts per square meter (W/m²), indicates the amount of energy a wave carries per unit time across a unit area.

Dimension of Intensity

The Intensity dimensional formula is: [M¹L⁰T^-3]

Derivation of Intensity Formula

When a point source emits energy uniformly in all directions, forming a spherical wave, the intensity of the wave decreases as the distance from the source increases, following the inverse-square law. According to this law, the intensity diminishes proportionally to the square of the distance from the source.

To express this mathematically, we can use the law of conservation of energy, which states that the total power emitted by the source remains constant. The power (P) emitted by the source can be calculated by integrating the intensity vector (I) over a closed surface that encloses the source. This integral is represented as:

P = ∫I⋅dA

Here, P represents the radiated power, I is the intensity vector, and dA is a small element of a closed surface containing the source.

If the intensity is uniform, meaning |I| is constant across the surface, and we integrate it over a surface perpendicular to the intensity vector, such as a sphere centered around the point source, the equation simplifies to:

P=I⋅A_surface =I⋅4πr²

In this equation:

• P is the power emitted by the source,
• I represents the intensity at the surface of the sphere,
• r is the radius of the sphere, and Asurface denotes the surface area of the sphere.

From the above equation, we can derive:

I = P/A

​This equation shows that the emitted power is proportional to the product of intensity and surface area, divided by the square of the distance from the source.

Applications of Intensity Formula

The intensity formula holds significance across diverse domains, offering practical applications in:

Lighting Design: Professionals in architecture and lighting design rely on intensity calculations to meticulously plan lighting arrangements for various settings, including buildings, streets, and public areas.

Environmental Monitoring: Intensity measurements play a pivotal role in environmental monitoring efforts aimed at evaluating light pollution levels and their ecological repercussions. 

Safety Engineering: In safety engineering, intensity calculations are instrumental in designing and implementing warning systems such as emergency beacons, traffic signals, and hazard lights. 

Material Processing: Industries involved in laser cutting, welding, and other material processing applications heavily rely on intensity control to regulate the energy delivered by laser beams.

Solved Examples on Intensity Formula

Example 1: A light source emits radiation uniformly in all directions with a power of 50 W. Calculate the intensity of radiation at a distance of 2 meters from the source.

Solution:

Given:

Power of the light source, 

P=50W

Distance from the source, 

r=2m

Using the intensity formula:

I= P/4πr²

​Substituting the given values:

I= 50/4π×2²

I= 50/16π

​Approximating the value of π to 3.14:

I≈0.995W/m²

 Hence, the intensity of radiation at a distance of 2 meters from the source is approximately 

0.995W/m²

Example 2: A laser beam with an intensity of 1000 W/m² is incident on a surface with an area of 0.5 m². Calculate the power of the laser beam incident on the surface.

Solution:

Intensity of the laser beI=1000W/m²

Area of the surface, 

A=0.5m²

Using the formula:

P=I×A

Substituting the given values:

P=1000×0.5

P=500W

Hence, the power of the laser beam incident on the surface is 500W.

Example 3:A light bulb emits radiation uniformly in all directions with an intensity of 800 W/m². Calculate the total power emitted by the bulb if its surface area is 0.2 m².

Solution:

Given:

Intensity (I) = 800 W/m²

Surface Area (A) = 0.2 m²

Using the formula:

P=I×A

Substituting the given values:

P=800×0.2

P=160W