Problem 12 - University Physics with Modern Physics

Question

The electric-field amplitude near a certain radio transmitter is \(3.85 \times 10^{-3} \mathrm{V} / \mathrm{m}\) What is the amplitude of \(\overrightarrow{\boldsymbol{B}} ?\) How does this compare in magnitude with the earth's field?

Step-by-Step Solution

Answer

The amplitude of \(\overrightarrow{\boldsymbol{B}}\) is approximately \(1.28 \times 10^{-11}\, \text{T}\), which is much smaller than the Earth's magnetic field.

Step 1: Understand the Relationship Between E and B

In electromagnetic waves, the relationship between the electric field amplitude \(E\) and the magnetic field amplitude \(B\) is given by the equation \(E = cB\), where \(c\) is the speed of light, approximately \(3 \times 10^8 \text{ m/s}\). We'll use this equation to find the amplitude of \(B\).

Step 2: Solve for the Magnetic Field Amplitude B

Since we know that \(E = 3.85 \times 10^{-3} \text{ V/m}\) and \(c = 3 \times 10^8 \text{ m/s}\), we can rearrange the equation to solve for \(B:\)\[B = \frac{E}{c} = \frac{3.85 \times 10^{-3}}{3 \times 10^8} \approx 1.28 \times 10^{-11} \text{ T}\]

Step 3: Compare B with the Earth's Magnetic Field

The amplitude of the Earth's magnetic field is approximately \(1 \times 10^{-5} \text{ T}\). To compare, divide the calculated \(B\) by the Earth's magnetic field:\[\frac{1.28 \times 10^{-11}}{1 \times 10^{-5}} = 1.28 \times 10^{-6}\]This shows that the magnetic field amplitude from the radio transmitter is much smaller than the Earth's magnetic field.

Key Concepts

Electric Field Amplitude

The electric field amplitude, often denoted as \(E\), is a crucial parameter in the study of electromagnetic waves. It represents the peak value of the electric field strength in a wave and is measured in volts per meter (V/m). Understanding the amplitude of an electric field can help us determine the intensity of the wave.

In electromagnetic waves, the electric field amplitude is directly related to the energy transported by the wave. The greater the amplitude, the more energy is carried through a given area. This concept plays a significant role in applications ranging from radio transmission to light waves. When examining electromagnetic waves like the radio wave in our exercise, knowing the electric field amplitude allows us to find other values like the magnetic field amplitude, using the fundamental equations that relate these concepts.

Magnetic Field Amplitude

The magnetic field amplitude, represented as \(B\), describes the peak strength of the magnetic field in an electromagnetic wave. It is measured in teslas (T), which is a unit expressing the amount of magnetism.

To find the magnetic field amplitude, we utilize the relationship between the electric field and magnetic field amplitudes in an electromagnetic wave: \(E = cB\), where \(c\) is the speed of light. By rearranging this formula, we can solve for \(B\):

• This equation shows that when the electric field amplitude is known, the magnetic field amplitude can be easily calculated by dividing \(E\) by \(c\).
• In the previous steps, we used these relations to find that a radio wave's magnetic field amplitude is \(1.28 \times 10^{-11} \text{ T}\), much weaker compared to the Earth's magnetic field.

This highlights how electromagnetic waves with different field amplitudes can vary dramatically in strength, affecting their interaction with surrounding environments and materials.

Speed of Light Relation

The speed of light, denoted as \(c\), is a fundamental constant in physics, critical to the study of electromagnetic waves. It has a value of approximately \(3 \times 10^8 \text{ m/s}\). This constant links the electric field amplitude \(E\) to the magnetic field amplitude \(B\) through the relation \(E = cB\).

• The formula \(E = cB\) allows us to understand the propagation of electromagnetic waves, as it ties together their electric and magnetic components in a simple relationship.
• This relationship indicates that for a given frequency, adjusting the electric field amplitude will proportionally adjust the magnetic field amplitude and vice versa.
• In practical applications, knowing the speed of light helps calculate other parameters such as intensity and wavelength of the electromagnetic waves, useful in technologies like communications and medical imaging.

Understanding this relation allows students to explore how changes in one field affect the other, providing deep insights into the nature and behavior of light and other electromagnetic phenomena.