Problem 2158 - JEE Physics

Question

Relation between amplitudes of electric and Magnetic field is (A) \(E_{0}=B_{0}\) (B) \(E_{0}=\mathrm{cB}_{0}\) (C) \(E_{0}=\left(B_{0} / c\right)\) (D) \(E_{0}=\left(\mathrm{c} / \mathrm{B}_{0}\right)\)

Step-by-Step Solution

Answer

The correct relationship between the amplitudes of electric and magnetic fields in an electromagnetic wave is: \(E_{0} = cB_{0}\)

Step 1: Recall the relationship between Electric and Magnetic fields in an electromagnetic wave

In an electromagnetic wave, the electric field (E) and the magnetic field (B) are perpendicular to each other and vary sinusoidally in both space and time. The relationship between the magnitudes of electric and magnetic fields in an electromagnetic wave is given by: \[E = cB\] where E is the magnitude of the electric field, B is the magnitude of the magnetic field, and c is the speed of light in a vacuum (\(3 \times 10^8\) m/s).

Step 2: Compare the equation with the given options

Now that we have the relationship between the magnitudes of electric and magnetic fields, let's compare it with the given options: (A) \(E_{0}=B_{0}\): This option implies that the amplitudes of the electric and magnetic fields are equal. However, it does not take into account the speed of light. (B) \(E_{0}=cB_{0}\): This option correctly states the relationship between the amplitudes of electric and magnetic fields, with the speed of light included. Based on our earlier equation, this option is correct. (C) \(E_{0}=(B_{0} / c)\): This option incorrectly implies that the amplitude of the electric field is equal to the amplitude of the magnetic field divided by the speed of light. (D) \(E_{0}=(c / B_{0})\): This option incorrectly implies that the amplitude of the electric field is equal to the speed of light divided by the amplitude of the magnetic field.

Step 3: Choose the correct option

Based on our comparison, the correct relationship between the amplitudes of electric and magnetic fields in an electromagnetic wave is Option (B): \(E_{0} = cB_{0}\)

Key Concepts

Electric Field

The electric field is a fundamental concept in physics that represents the influence an electric charge exerts on its surroundings. Imagine it as an invisible field that emanates from electric charges, affecting other charges within its realm. In electromagnetic waves, which combine both electric and magnetic components, the electric field oscillates perpendicularly to the magnetic field, together creating wave-like behavior.

Key points about electric fields in electromagnetic waves include:

• The electric field ( \( E \) ) in these waves is sinusoidal, meaning it varies periodically both in time and space.
• It is perpendicular to the magnetic field, meaning they are at right angles to each other.
• The magnitude of the electric field is proportional to the magnetic field, interconnected by the speed of light ( \( c \) ), as per the equation \( E = cB \) .

In everyday terms, think of the electric field as the component of the wave that can exert forces on charges, causing them to move, which underlays phenomena such as radio waves transmitting signals from a tower to your car radio.

Magnetic Field

Like the electric field, the magnetic field is another invisible influence that emanates from moving charges. Within electromagnetic waves, the magnetic field works in tandem with the electric field. You can envision it as another wave running alongside the electric component, perpendicular to it.

Some important aspects of the magnetic field in electromagnetic waves include:

• The magnetic field ( \( B \) ) is also sinusoidal, mirroring the periodic nature of the electric field.
• It is perpendicular not only to the electric field but also to the direction of wave propagation.
• The strength of the magnetic field is proportionally linked to the electric field, reflected in the equation \( E = cB \) .

These fields together embody the flowing and propagating nature of electromagnetic waves, such as light, that travel through space, allowing for communication and transfer of energy.

Speed of Light

The speed of light, symbolized by \( c \) , plays a pivotal role in the relationship between the electric and magnetic fields in electromagnetic waves. It is a universal constant, approximately \( 3 \times 10^8 \text{ m/s} \) , the fastest speed at which information or matter can travel.

Here are some key elements about the speed of light:

• It dictates how quickly electromagnetic waves propagate through a vacuum.
• In the equation \( E = cB \) , it represents the proportionality factor that links electric and magnetic fields, indicating that higher magnetic fields correspond to stronger electric fields.
• It remains constant and is not influenced by the relative motion of the source or observer, showcasing its supreme consistency across physics.

Understanding the speed of light not only helps conceptualize electromagnetic phenomena but also underpins technologies and theories, from GPS to Einstein’s theory of relativity, highlighting the fundamental nature of this constant in our universe.