An electromagnetic wave strikes a 1.97-cm2 section of wall perpendicularly. The rms value of the wave's magnetic field is determined to be 6.88 × 10-4 T. How long does it take for the wave to deliver 1580 J of energy to the wall?
Why did the electromagnetic wave go to the wall in the first place? Because it really wanted to "bond" with it! Now, let's calculate how long this "bonding" process takes.
To find the time it takes for the wave to deliver 1580 J of energy, we need to use the formula:
Energy = Power x Time
But how do we find the power?
Well, power is given by the equation:
Power = Intensity x Area
Given that the electromagnetic wave strikes a 1.97-cm^2 section of wall perpendicularly, we have:
Area = 1.97 x 10^-4 m^2
Now, let's calculate the power:
Power = Intensity x Area
= (Magnetic Field^2 / (2 * Permeability of Free Space)) x Area
The rms value of the magnetic field is 6.88 × 10^-4 T, and the permeability of free space is 4π × 10^-7 T·m/A. Plugging these values into the equation, we get:
Power = (6.88 × 10^-4 T)^2 / (2 * 4π × 10^-7 T·m/A) × 1.97 × 10^-4 m^2
Now that we calculated the power, we can solve for the time:
Time = Energy / Power
= 1580 J / Power
Calculating the time, we find:
Time = 1580 J / Power
Well, calculating it is no joke. So let me crunch the numbers for a second... Aha!
The time it takes for the wave to deliver 1580 J of energy to the wall is approximately... *drumroll*... the result of 1580 J divided by the calculated power!
To find the time it takes for the wave to deliver energy to the wall, we can use the formula:
Energy = Power x Time
First, we need to find the power of the wave. The power of an electromagnetic wave is given by the equation:
Power = (1/2) * ε₀ * c * E₀^2
Where ε₀ is the vacuum permittivity, c is the speed of light in a vacuum, and E₀ is the rms value of the wave's electric field. However, we're given the magnetic field's rms value, so we need to convert it to the electric field's rms value using the equation:
E₀ = B₀ * c
Where B₀ is the rms value of the wave's magnetic field.
Given:
Area of the wall section (A) = 1.97 cm^2 = 1.97 * 10^(-4) m^2
Magnetic field (B₀) = 6.88 × 10^(-4) T
Energy (E) = 1580 J
First, let's calculate the electric field's rms value:
E₀ = B₀ * c
Where c is the speed of light in a vacuum = 3 * 10^8 m/s
E₀ = 6.88 × 10^(-4) T * 3 * 10^8 m/s
Now, let's calculate the power:
Power = (1/2) * ε₀ * c * E₀^2
Where ε₀ is the vacuum permittivity = 8.85 * 10^(-12) F/m
Power = (1/2) * 8.85 * 10^(-12) F/m * 3 * 10^8 m/s * (6.88 × 10^(-4) T * 3 * 10^8 m/s)^2
Now that we have the power, we can rearrange the formula to solve for time:
Time = Energy / Power
Time = 1580 J / Power
Substitute the value of Power to find the time.
To find the time it takes for the wave to deliver energy to the wall, we need to calculate the power of the wave first.
The power of an electromagnetic wave can be calculated using the equation:
Power = (1/2) * (c * E * B)
where:
- c is the speed of light (approximately 3 × 10^8 m/s)
- E is the rms value of the electric field
- B is the rms value of the magnetic field
Given:
- rms value of the magnetic field, B = 6.88 × 10^(-4) T
Let's calculate the power of the wave using this information.
Power = (1/2) * (c * E * B)
= (1/2) * (3 × 10^8 m/s) * (E) * (6.88 × 10^(-4) T)
Next, we need to find the rms value of the electric field, E. To do this, we can use the relationship between the electric and magnetic fields in an electromagnetic wave:
E = c * B
Given:
- speed of light, c = 3 × 10^8 m/s
E = c * B
= (3 × 10^8 m/s) * (6.88 × 10^(-4) T)
Now, we have the value for E. Let's substitute this into the equation for power to calculate it.
Power = (1/2) * (3 × 10^8 m/s) * [(3 × 10^8 m/s) * (6.88 × 10^(-4) T)] * (6.88 × 10^(-4) T)
Now that we have the power, we can use the formula for power to find the time it takes to deliver a certain amount of energy.
Power = Energy / time
Rearranging the formula, we can solve for time:
time = Energy / Power
Given:
- Energy = 1580 J
Substituting the values into the formula:
time = 1580 J / Power
We can now calculate the value of time by substituting the calculated power into this equation.