A physical quantity $\vec{ S }$ is defined as $\vec{ S }=(\vec{ E } \times \vec{ B }) / \mu_{0}$, where $\vec{ E }$ is electric field, $\vec{ B }$ is magnetic field and $\mu_{0}$ is the permeability of free space. The dimensions of $\vec{ S }$ are the same as the dimensions of which of the following quantity(ies)?
- $\frac{\text { Energy }}{\text { Charge } \times \text { Current }}$
- $\frac{\text { Force }}{\text { Lenght } \times \text { Time }}$
- $\frac{\text { Energy }}{\text { Volume }}$
- $\frac{\text { Power }}{\text { Area }}$
The Correct Option is D
Solution and Explanation
The given physical quantity is defined as:
\( \vec{S} = \frac{\vec{E} \times \vec{B}}{\mu_0} \)
Where \( \vec{E} \) is the electric field, \( \vec{B} \) is the magnetic field, and \( \mu_0 \) is the permeability of free space. We are asked to find the dimensions of \( \vec{S} \) and compare it to the dimensions of other physical quantities.
Step 1: Analyzing the dimensions of each term
The electric field \( \vec{E} \) has dimensions of force per unit charge, which can be expressed as:
\( [E] = \frac{ML}{T^3I} \)
The magnetic field \( \vec{B} \) has dimensions of force per unit charge per velocity, which can be written as:
\( [B] = \frac{M}{T^2I} \)
The permeability of free space \( \mu_0 \) has dimensions:
\( [\mu_0] = \frac{M}{A^2T^2} \)
Step 2: Calculating the dimensions of \( \vec{S} \)
The dimensions of \( \vec{S} \) are given by:
\( [\vec{S}] = \frac{[E] \times [B]}{[\mu_0]} \)
Substituting the dimensions of \( \vec{E} \), \( \vec{B} \), and \( \mu_0 \):
\( [\vec{S}] = \frac{\left( \frac{ML}{T^3I} \right) \times \left( \frac{M}{T^2I} \right)}{\frac{M}{A^2T^2}} = \frac{M^2L}{T^5I^2} \times \frac{A^2T^2}{M} = \frac{ML^2}{T^3I^2A^2} \)
Step 3: Identifying the quantity with the same dimensions
The dimensions of \( \vec{S} \) are the same as the dimensions of:
\( \frac{\text{Power}}{\text{Area}} \)
This is because power has dimensions of \( \frac{ML^2}{T^3I} \) and area has dimensions of \( L^2 \), so \( \frac{\text{Power}}{\text{Area}} \) has the same dimensions as \( \vec{S} \).
Final Answer:
\( \frac{\text{Power}}{\text{Area}} \)
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Concepts Used:
Electromagnetic waves
The waves that are produced when an electric field comes into contact with a magnetic field are known as Electromagnetic Waves or EM waves. The constitution of an oscillating magnetic field and electric fields gives rise to electromagnetic waves.
Types of Electromagnetic Waves:
Electromagnetic waves can be grouped according to the direction of disturbance in them and according to the range of their frequency. Recall that a wave transfers energy from one point to another point in space. That means there are two things going on: the disturbance that defines a wave, and the propagation of wave. In this context the waves are grouped into the following two categories:
- Longitudinal waves: A wave is called a longitudinal wave when the disturbances in the wave are parallel to the direction of propagation of the wave. For example, sound waves are longitudinal waves because the change of pressure occurs parallel to the direction of wave propagation.
- Transverse waves: A wave is called a transverse wave when the disturbances in the wave are perpendicular (at right angles) to the direction of propagation of the wave.