Intensity Formula: Derivation and Solved Examples

The intensity formula measures the power per unit area for various types of energy that propagate as waves.

• The intensity of a wave is the amount of energy it transmits per unit time across a unit area of a surface.
• It is a measure of the energy transmitted by a wave.
• It is also known flux of radiant energy.
• The unit of measurement of intensity is watt per squared meter (W/m²).
• The intensity of a wave is determined by its strength and amplitude.
• Intensity is represented by the letter \(I\).

Key Terms: Wave Intensity, Sound Intensity, Acoustic Intensity, Luminous Intensity, Amplitude, Frequency, Energy, Power, Law of conservation of energy

Intensity

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The power transferred per unit area is known as the intensity or flux of radiant energy.

• It is denoted as ‘\(I\)’.
• The area is measured on a plane perpendicular to the direction of energy propagation.
• The SI unit of intensity is watts per square meter (W/m²), or kg s⁻³ in base units.
• The dimensional formula of intensity is [M T^-3].
• It is a scalar quantity.

The intensity of a wave is the quantity of energy that the wave conveys per unit time across the surface of the unit area, and it is also equivalent to the energy density multiplied by the wave speed. 

• The magnitude of intensity will depend on the strength and amplitude of a wave under propagation.
• Luminous Intensity or Light Intensity is the amount of visible light emitted from the source/ brightness of light from the source per unit distance.
• Sound Intensity can be calculated as the power carried by the sound waves per unit area in the direction perpendicular to that area.
• It is also known as Acoustic Intensity.
• The minimum intensity of sound that a human can hear is 10^-12​ W m^-2 and the maximum is 1 W m^-2.

Intensity Formula

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The formula of intensity is given by

\(I=\frac {P}{A}\)

Where

• I is the intensity
• P is the power
• A is the area of cross-section​

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Derivation of Intensity Formula

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If a point source emits energy in all directions (creating a spherical wave), and no energy is absorbed or scattered by the medium, the intensity decreases proportional to the distance from the object squared. This is an example of the inverse-square law.

From the law of conservation of energy, considering the net power emanating is constant,

\(P = \int I.dA\)

Where

• P is the radiated power
• I is the intensity vector
• dA is a small element of a closed surface that contains the source.

When a uniform intensity, |I| = constant, is integrated across a surface perpendicular to the intensity vector, such as a sphere centered around the point source, the equation becomes

\(P=I.A_{surface}=I. 4\pi r^2\)

Where

• I is the intensity at the sphere’s surface
• r is the radius of the sphere
• A_surface is the surface area of the sphere

From the above equation, we get

\(P=\frac {I}{A} = \frac {P}{4\pi r^2}\)

Solved Examples

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Ques. Calculate the intensity of a wave whose power is 20 KW, and the area of cross-section is 20 x 10⁶ m².

• Power of the wave, P = 20 KW = 20 ×10³ W
• Area of the cross-section, A = 20 ×10⁶ m²

The formula of intensity is given by

I = P / A

On substituting the values, we get

I = (20 × 10³)/(20 × 10⁶)

⇒ I = 10^-2 W/m²

Ques. What is the intensity of light incident normal to a circular surface of radius 4 cm from a 200 W source of light?

• Radius of the circular surface, r = 4 cm = 0.04 m,
• Power of the source of light, P = 200 W

The area of the circular surface is given by

A = πr²

⇒ A = 3.14 x (0.04)² = 0.005 m²

The intensity of incident light is given by

I = P/A

On substituting the values, we get

I = 200/0.005

⇒ I = 4 x 10³ W/m²

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Things to Remember

• Intensity is defined as the power delivered per unit area of a surface.
• The formula of intensity is given as I = P/A.
• Intensity is a scalar quantity.
• The dimensional formula of intensity is [M T^-3].
• SI unit of intensity is W/m².
• The minimum intensity of sound that a human can hear is 10^-12 W m^-2 and the maximum is 1 W m^-2.