The electromagnetic wave that delivers a cellular phone call to a car has a magnetic field with an rms value of 1.4 × 10-10 T. The wave passes perpendicularly through an open window, the area of which is 0.24 m2. How much energy does this wave carry through the window during a 48-s phone call? The electromagnetic wave that delivers a cellular phone call to a car has a magnetic field with an rms value of 1.4 × 10-10 T. The wave passes perpendicularly through an open window, the area of which is 0.24 m2. How much energy does this wave carry through the window during a 48-s phone call?
The Correct Answer and Explanation is :
The energy carried by an electromagnetic wave passing through an area is related to the intensity of the wave, which can be determined using the magnetic field strength and the area through which the wave passes. Let’s break down the problem and solve it step by step.
Given:
- RMS magnetic field strength (Bₓrms): ( 1.4 \times 10^{-10} \, \text{T} )
- Area of the window (A): ( 0.24 \, \text{m}^2 )
- Time duration (t): ( 48 \, \text{s} )
Step 1: Calculate the intensity of the electromagnetic wave
The intensity ( I ) of an electromagnetic wave is related to the RMS value of the magnetic field ( B_\text{rms} ) by the following equation:
[
I = \frac{c B_\text{rms}^2}{2 \mu_0}
]
Where:
- ( I ) is the intensity of the wave (energy per unit area per unit time),
- ( c ) is the speed of light in a vacuum ( (3.0 \times 10^8 \, \text{m/s}) ),
- ( B_\text{rms} ) is the RMS value of the magnetic field,
- ( \mu_0 ) is the permeability of free space ( (4 \pi \times 10^{-7} \, \text{T·m/A}) ).
Step 2: Energy through the window
The energy ( E ) passing through the area ( A ) during the time ( t ) is:
[
E = I \times A \times t
]
Now, let’s plug in the given values to calculate the energy.
Let me calculate this now:
The energy carried by the electromagnetic wave through the window during the 48-second phone call is approximately ( 2.7 \times 10^{-5} \, \text{J} ).
Explanation:
- Intensity of the Wave: The intensity of the wave is related to the RMS value of the magnetic field, which reflects the strength of the magnetic component of the electromagnetic wave. The formula for intensity uses both the speed of light and the magnetic field strength, indicating how much energy the wave transports through a unit area per unit of time.
- Energy Through the Window: The energy that passes through the window is determined by the intensity multiplied by the area the wave passes through and the duration of the event (the phone call). This gives us the total energy the wave carries through the window.
This is how we calculate the energy transmitted through the window in the given situation.
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