Question

The maximum electric field in a plane electromagnetic wave is \(900 \mathrm{NC}^{-1}\). The wave is going in the \(\mathrm{x}\) direction and the electric field is in the y direction. The maximum magnetic field in the wave is \(\mathrm{T}\) (A) \(3 \times 10^{-8}\) (B) \(3 \overline{\times 10^{-6}}\) (C) \(27 \times 10^{-6}\) (D) \(27 \times 10^{10}\)

Short answer

The maximum magnetic field (\(B_{max}\)) in the plane electromagnetic wave is \(3 \times 10^{-6} T\). This is found by using the formula \(E_{max} = c \cdot B_{max}\), where \(E_{max}\) is the maximum electric field (900 N/C) and c is the speed of light in a vacuum (\(3 \times 10^8 m/s\)). Solving for \(B_{max}\), we get the correct option (B) \(3 \times 10^{-6} T\).

Step-by-step solution

Write down the given information

We are given the following information: - Maximum electric field (\(E_{max}\)) = 900 N/C - Electromagnetic wave is traveling in the x direction - Electric field is in the y direction - We need to find the maximum magnetic field (\(B_{max}\))

Recall the formula relating the electric and magnetic fields in an electromagnetic wave

In an electromagnetic wave, the maximum electric field (\(E_{max}\)) and maximum magnetic field (\(B_{max}\)) are related by the following formula: \(E_{max} = c \cdot B_{max}\) where c is the speed of light in a vacuum, approximately \(3 \times 10^8 m/s\).

Solve for the maximum magnetic field

We can find the maximum magnetic field (\(B_{max}\)) by rearranging the equation found in step 2: \(B_{max} = \frac{E_{max}}{c}\) Now, substitute the given value of the maximum electric field and the speed of light to solve for the maximum magnetic field: \(B_{max} = \frac{900 N/C}{3 \times 10^8 m/s}\) \(B_{max} = 3 \times 10^{-6} T\)

Choose the correct option

Since the maximum magnetic field (\(B_{max}\)) is \(3 \times 10^{-6} T\), the correct option is (B) \(3 \times 10^{-6} T\).

Key concepts

Electric Field

The electric field is a crucial component of an electromagnetic wave. Imagine it's like an invisible force field. It creates a force on other charges. The strength of this field determines how much force it can apply. In the given exercise, the maximum electric field is 900 N/C. This means when the wave passes through space, it can push or pull a charge with a force of 900 Newtons per Coulomb.

This electric field oscillates at right angles to the direction of the wave. So, in our exercise, the wave travels in the x direction, which means the electric field aligns along the y direction. This perpendicular alignment is a key feature of electromagnetic waves, always working hand in hand with the magnetic field to propagate the wave.

Magnetic Field

The magnetic field in an electromagnetic wave works alongside the electric field. Similar to the electric field, the magnetic field is also perpendicular to the wave's direction of travel.

In our scenario, the magnetic field would be in the z direction if the electric field is in the y direction, creating a right-hand rule arrangement with the wave's direction in the x axis. The magnetic field isn't as strong as the electric field, but it is equally essential for the propagation of the wave. In electromagnetic waves, these two fields are directly related by the speed of light. The given problem showed that the maximum value of the magnetic field is calculated as \( B_{max} = 3 \times 10^{-6} \) T, by using this relationship.

Speed of Light

The speed of light, often labeled as \( c \), is approximately \( 3 \times 10^8 \) meters per second in a vacuum. It's not only a measure of how fast light travels but also a critical constant in physics.

In electromagnetic waves, the electric and magnetic fields oscillate in a way that their product is equal to the speed of light. This is why we use the formula \( E_{max} = c \cdot B_{max} \) in problems involving electromagnetic waves.

• The speed of light connects electric and magnetic fields.
• It governs the behavior of waves in free space, guiding how they travel and interact.
• Beyond its role in electromagnetism, it is crucial in theories of relativity.

Understanding the speed of light is vital because it helps to unravel the relationship between the two fields and facilitates calculations in physics problems across different scenarios.