A source of electromagnetic waves radiates with power P uniformly in all directions. At some distance d from the source, the magnetic field amplitude of the waves is Bo. What will the amplitude of the magnetic field be a distance d/4 away from the source, in terms of Bo, P, d, to, µo, and/or c? Justify your answer.
A source of electromagnetic waves radiates with power P uniformly in all directions. At some distance d from the source, the magnetic field amplitude of the waves is Bo. What will the amplitude of the magnetic field be a distance d/4 away from the source, in terms of Bo, P, d, to, µo, and/or c? Justify your answer.
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![### Problem Statement
**Question:**
A source of electromagnetic waves radiates with power \( P \) uniformly in all directions. At some distance \( d \) from the source, the magnetic field amplitude of the waves is \( B_0 \). What will the amplitude of the magnetic field be a distance \( d/4 \) away from the source, in terms of \( B_0, P, d, \epsilon_0, \mu_0, \) and/or \( c \)? Justify your answer.
**Text Explanation:**
In this problem, we are asked to determine the amplitude of the magnetic field at a distance \( d/4 \) from an electromagnetic wave source, given that the amplitude at a distance \( d \) is \( B_0 \). The source radiates uniformly in all directions with power \( P \). The variables provided for deriving the solution include \( B_0 \) (magnetic field amplitude at distance \( d \)), \( P \) (power of the source), \( d \) (initial distance), \( \epsilon_0 \) (permittivity of free space), \( \mu_0 \) (permeability of free space), and \( c \) (speed of light in vacuum).
We'll need to understand the relationship between the magnetic field amplitude and the distance from the source, leveraging principles from electromagnetism and wave propagation.
### Solution Outline
1. **Inverse Square Law**:
- Electromagnetic wave intensity falls off as the square of the distance from the source. Thus, if the distance is reduced, the power density increases proportionally.
2. **Intensity and Magnetic Field Relation**:
- The intensity (\(I\)) of the electromagnetic wave is proportional to the square of the magnetic field amplitude (\(B\)).
3. **Calculations**:
\[
I \propto \left(\frac{1}{r^2}\right) \quad \text{and} \quad I \propto B^2
\]
Thus,
\[
B^2 \propto \frac{1}{d^2}
\]
4. **Given Values and Distance Change**:
At distance \( d \):
\[
B_0^2 \propto \frac{1}{d^2}
\]
At distance \( \frac{d}{4}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F651b911d-2c53-4f54-b4b6-29cb6ac234d2%2Ffdc63d48-0e0e-4487-b173-501dddc79021%2F9q4guck_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Question:**
A source of electromagnetic waves radiates with power \( P \) uniformly in all directions. At some distance \( d \) from the source, the magnetic field amplitude of the waves is \( B_0 \). What will the amplitude of the magnetic field be a distance \( d/4 \) away from the source, in terms of \( B_0, P, d, \epsilon_0, \mu_0, \) and/or \( c \)? Justify your answer.
**Text Explanation:**
In this problem, we are asked to determine the amplitude of the magnetic field at a distance \( d/4 \) from an electromagnetic wave source, given that the amplitude at a distance \( d \) is \( B_0 \). The source radiates uniformly in all directions with power \( P \). The variables provided for deriving the solution include \( B_0 \) (magnetic field amplitude at distance \( d \)), \( P \) (power of the source), \( d \) (initial distance), \( \epsilon_0 \) (permittivity of free space), \( \mu_0 \) (permeability of free space), and \( c \) (speed of light in vacuum).
We'll need to understand the relationship between the magnetic field amplitude and the distance from the source, leveraging principles from electromagnetism and wave propagation.
### Solution Outline
1. **Inverse Square Law**:
- Electromagnetic wave intensity falls off as the square of the distance from the source. Thus, if the distance is reduced, the power density increases proportionally.
2. **Intensity and Magnetic Field Relation**:
- The intensity (\(I\)) of the electromagnetic wave is proportional to the square of the magnetic field amplitude (\(B\)).
3. **Calculations**:
\[
I \propto \left(\frac{1}{r^2}\right) \quad \text{and} \quad I \propto B^2
\]
Thus,
\[
B^2 \propto \frac{1}{d^2}
\]
4. **Given Values and Distance Change**:
At distance \( d \):
\[
B_0^2 \propto \frac{1}{d^2}
\]
At distance \( \frac{d}{4}
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