Question

The maximum value of \(\mathrm{E}^{-}\) in an electromagnetic waves in air is equal to \(6.0 \times 10^{-4} \mathrm{Vm}^{-1}\). The maximum value of \(\mathrm{B}^{-}\) is (A) \(1.8 \times 10^{5} \mathrm{~T}\) (B) \(2.0 \times 10^{4} \mathrm{~T}\) (C) \(2.0 \times 10^{-12} \mathrm{~T}\) (D) \(1.8 \times 10^{13} \mathrm{~T}\)

Short answer

The maximum value of the magnetic field (B) in an electromagnetic wave is (C) \(2.0 \times 10^{-12} T\).

Step-by-step solution

Write down the given values

We are given the maximum value of the electric field (E) to be: \(E_{max} = 6 \times 10^{-4} Vm^{-1}\) And we know the speed of light (c) is: \(c = 3 \times 10^8 ms^{-1}\)

Calculate the maximum value of B

Use the formula for the magnetic field: \(B_{max} = \frac{E_{max}}{c}\) Plug in the given values: \(B_{max} = \frac{6 \times 10^{-4} Vm^{-1}}{3 \times 10^8 ms^{-1}}\) Now, simplify the equation: \(B_{max} = 2 \times 10^{-12} T\)

Compare the result with the given options

Now let's compare our result with the given options: (A) \(1.8 \times 10^{5} T\) (B) \(2.0 \times 10^{4} T\) (C) \(2.0 \times 10^{-12} T\) (D) \(1.8 \times 10^{13} T\) We see that our calculated value of \(B_{max}\) matches option (C). Therefore, the correct answer is (C) \(2.0 \times 10^{-12} T\).

Key concepts

Electric Field

The **electric field**, often denoted as \( E \), is a fundamental concept in electromagnetism. It represents the force per unit charge exerted on a stationary charge in an electric field and is measured in volts per meter (Vm^-1). The electric field is a vector, meaning it has both magnitude and direction.

In the context of electromagnetic waves, such as light or radio waves, the electric field oscillates in a direction perpendicular to the wave's direction of propagation. This oscillation is crucial because it helps in transmitting energy through space. Electromagnetic waves consist of both electric and magnetic fields oscillating perpendicular to each other and to the direction of wave propagation.

An electric field in an electromagnetic wave can have a maximum value, known as the amplitude. This "maximum value of the electric field" indicates the peak strength of the electric force experienced by a charge within that wave. For example, in our given exercise, the maximum value of \( E \) is \( 6.0 \times 10^{-4} \) V/m.

Magnetic Field

The **magnetic field**, represented as \( B \), is an essential part of electromagnetic waves. It is usually measured in teslas (T). Like the electric field, the magnetic field in an electromagnetic wave is a vector and oscillates perpendicular to the direction of the wave as well as the electric field.

A fascinating aspect of electromagnetic waves is that when the electric field is at its maximum, the magnetic field also reaches a maximum, but because of the nature of their relationship, these maximums do not occur in phase (they reach their peaks at different times along the same wave). These fields are interdependent; the changing electric field produces the magnetic field, and vice versa.

In practical calculations, the maximum value of the magnetic field can be found using the formula \( B_{max} = \frac{E_{max}}{c} \), where \( c \) is the speed of light. In the exercise provided, we computed \( B_{max} \) to be \( 2 \times 10^{-12} \) T, matching the correct answer as option (C). Understanding this relationship between the electric and magnetic fields allows us to predict how electromagnetic waves will behave as they travel through different mediums.

Speed of Light

The **speed of light**, denoted by \( c \), is a constant value crucial to understanding electromagnetic waves. In a vacuum, this speed is approximately \( 3 \times 10^8 \) meters per second (ms^-1).

This constant isn't just a speed; it's a cornerstone constant in physics, influencing equations governing electromagnetic interactions. Notably, \( c \) relates the maximum values of electric and magnetic fields in an electromagnetic wave. The relationships derived from this constant are fundamental, forming the basis of theories like Einstein’s theory of relativity. In (many problem-solving) contexts of electromagnetic waves, multiplying or dividing by \( c \) allows us to interconvert metrics between electric and magnetic fields.

In our exercise, the relationship \( B_{max} = \frac{E_{max}}{c} \) showcases how the speed of light ties together these fields' maximum values. Hence, knowing the speed of light allows us to understand and utilize formulas efficiently to analyze and predict wave behaviors.