You want to get an idea of the magnitude of magnetic fields produced by overhead power lines. You estimate that a transmission wire is about 11m above the ground. The local power company tells you that the line operates at 13kV and provide a maximum of 86MW to the local area.
Estimate the maximum magnetic field you might experience walking under such a power line, and compare to the Earth's field. [For an ac current, values are rms, and the magnetic field will be changing.]
To estimate the maximum magnetic field produced by the overhead power line, we need a few pieces of information: the voltage of the line, the distance from the line, and the maximum power it can provide.
First, let's consider the voltage of the line. The power company has provided the information that the line operates at 13 kilovolts (13 kV). This voltage represents the potential difference across the line.
Next, we need to calculate the electric current flowing through the line. To do so, we can use the power provided. The maximum power the line can deliver is given as 86 megawatts (86 MW). Since power is the product of current and voltage (P = IV), we can rearrange the formula to solve for the current (I):
I = P / V,
where I is the current, P is the power, and V is the voltage.
Plugging in the values, we have:
I = 86 MW / 13 kV.
To simplify the calculation, we need to convert the power to watts and the voltage to volts:
86 MW = 86,000,000 W,
13 kV = 13,000 V.
Now we can calculate the current:
I = 86,000,000 W / 13,000 V.
The next step is to estimate the maximum magnetic field experienced while walking under the power line. To do this, we can use the formula:
B = (μ0 * I) / (2 * π * r),
where B is the magnetic field, μ0 is the permeability of free space (4π x 10^-7 T·m/A), I is the current in amperes, and r is the distance from the wire.
Given that the transmission wire is about 11 m above the ground, we can assume that the distance from the wire (r) is equal to 11 m.
Substituting the values into the formula, we have:
B = (4π x 10^-7 T·m/A) * I / (2 * π * 11 m).
Simplifying the formula:
B = (2 x 10^-7 T·m/A) * I / 11 m.
Now, we can calculate the maximum magnetic field:
B = (2 x 10^-7 T·m/A) * (86,000,000 W / 13,000 V) / 11 m.
Calculating this expression will give us the estimated maximum magnetic field produced by the power line.
Finally, we can compare this value to the Earth's magnetic field, which is approximately 25 to 65 microteslas (μT). By comparing the magnitudes, we can determine how the power line's magnetic field compares to the Earth's field.