Problem 2149 - JEE Physics

Question

In an electromagnetic wave, if the amplitude of magnetic field is \(3 \times 10^{-10} \mathrm{~T}\), the amplitude of the associated electric field will be (A) \(9 \times 10^{-2} \overline{\mathrm{Vm}^{-1}}\) (B) \(3 \times 10^{-10} \mathrm{Vm}^{-1}\) (C) \(3 \times 10^{-2} \mathrm{Vm}^{-1}\) (D) \(1 \times 10^{-18} \mathrm{Vm}^{-1}\)

Step-by-Step Solution

Answer

The amplitude of the associated electric field is \(9 \times 10^{-2} \mathrm{Vm}^{-1}\) (Option A).

Step 1: Write down the given values

The amplitude of the magnetic field is given as \(B = 3 \times 10^{-10} T\).

Step 2: Write down the relationship between electric and magnetic field amplitudes

The relationship between electric and magnetic field amplitudes is given by the formula: \(E = cB\), where \(E\) is the amplitude of the associated electric field, \(B\) is the amplitude of the magnetic field, and \(c\) is the speed of light in a vacuum (\(3\times 10^8 m/s\)).

Step 3: Substitute the given values into the formula

Substitute the given value of the magnetic field amplitude \(B = 3 \times 10^{-10} T\) and the speed of light \(c = 3 \times 10^8 m/s\) into the formula: \(E = (3 \times 10^8 m/s)(3 \times 10^{-10} T)\)

Step 4: Calculate the electric field amplitude

Multiply the values in the expression: \(E = 9 \times 10^{-2} Vm^{-1}\)

Step 5: Compare the result with the given options

Comparing the result \(E = 9 \times 10^{-2} Vm^{-1}\) with the given options, we see that it matches option (A). The amplitude of the associated electric field is \(9 \times 10^{-2} Vm^{-1}\) (Option A).

Key Concepts

Amplitude of Magnetic Field

The amplitude of a magnetic field in an electromagnetic wave is an important aspect of understanding how these waves behave and propagate. In simple terms, the amplitude of a magnetic field represents its maximum strength at any given point. This affects the intensity of the wave, similar to how brightness works for light waves.

The unit of measurement for magnetic field amplitude is Tesla (T). A typical example, like the one given in the problem, shows an amplitude of a magnetic field as small as \(3 \times 10^{-10} \ \mathrm{T}\). Such measurements demonstrate how even extremely weak fields are part of our natural surroundings.

Understanding the amplitude can tell us:

• How strong the field is at its maximum.
• The potential energy within the wave.
• The interaction potential with materials that possess magnetic properties.

By considering these factors, scientists can make predictions about how electromagnetic waves will behave under different conditions.

Amplitude of Electric Field

In electromagnetic waves, the amplitude of the electric field is a key variable that defines the wave’s strength. Much like how the amplitude of the magnetic field highlights its strength, electric field amplitude is crucial as it affects the force exerted on charges within the field.

The relationship between the electric field amplitude (\(E\)) and the magnetic field amplitude (\(B\)) is determined by the speed of light (\(c\)) in a vacuum. The equation \(E = cB\) is used to calculate the electric field’s amplitude, where \(c = 3 \times 10^8 \ \mathrm{m/s}\).

For example, when calculating from the given magnetic field amplitude of \(3 \times 10^{-10} \ \mathrm{T}\), the electric field amplitude will be:

• \(E = (3 \times 10^8 \ \mathrm{m/s})(3 \times 10^{-10} \ \mathrm{T}) \)
• \(E = 9 \times 10^{-2} \ \mathrm{Vm^{-1}}\)

This calculation shows the capacity of the electric field to influence charged particles and its role in the wave's energy dynamics.

Speed of Light

The speed of light (\(c\)) in a vacuum is a universal constant pivotal to the physics of electromagnetic waves. It is approximately \(3 \times 10^8 \ \mathrm{m/s}\). This speed is not only crucial for understanding light but also electromagnetic radiation in general.

Here’s why the speed of light is significant:

• It is the fastest speed at which information or energy can travel through the vacuum of space.
• It links the electric and magnetic field amplitudes, as seen in the relationship \(E = cB\).
• It serves as a base for defining other physical quantities and constants in physics, such as the fine-structure constant.

The speed of light is fundamental in equations describing the propagation of electromagnetic waves, influencing both theoretical calculations and practical applications alike. Understanding this speed is crucial for fields ranging from telecommunications to theoretical physics.