The magnetic field at the center of a 0.70-cm-diameter loop is 2.4 mT.
What is the current in the loop?
A long straight wire carries the same current you found in part a. At what distance from the wire is the magnetic field 2.4 mT?
To find the current in the loop, we can use Ampere's Law and the formula for the magnetic field at the center of a loop.
Ampere's Law states that the line integral of the magnetic field around a closed loop is equal to the product of the current passing through the loop and the permeability of free space (µ₀).
For a loop of radius (r) and a magnetic field at its center (B), the formula for the magnetic field at the center is given by:
B = (µ₀ * I) / (2 * r)
Where:
B = Magnetic field at the center of the loop
I = Current in the loop
µ₀ = Permeability of free space
r = Radius of the loop
From the given information, we know that the magnetic field at the center of the loop (B) is 2.4 mT, which is equivalent to 0.0024 T. The radius of the loop (r) is given as 0.70 cm, which is equivalent to 0.007 m.
Now, we can rearrange the formula to solve for the current (I):
I = (B * 2 * r) / µ₀
Plugging in the values, we have:
I = (0.0024 T * 2 * 0.007 m) / µ₀
The value of the permeability of free space (µ₀) is 4π x 10^-7 T·m/A.
Substituting the value of µ₀, we get:
I = (0.0024 T * 2 * 0.007 m) / (2π x 10^-7 T·m/A)
Simplifying the equation, we find:
I ≈ 0.00857 A
Therefore, the current in the loop is approximately 0.00857 A.
To find the distance from a long straight wire where the magnetic field is 2.4 mT, we can use the formula for the magnetic field caused by a straight wire.
The formula to calculate the magnetic field (B) at a distance (d) from a long straight wire with current (I) is given by:
B = (µ₀ * I) / (2π * d)
Rearranging the formula to solve for the distance (d), we have:
d = (µ₀ * I) / (2π * B)
We already know the current (I) from part a, which is approximately 0.00857 A, and the desired magnetic field (B) is 2.4 mT or 0.0024 T.
Again, substituting the value of the permeability of free space (µ₀), which is 4π x 10^-7 T·m/A, we get:
d = (2π x 10^-7 T·m/A * 0.00857 A) / (2π * 0.0024 T)
Simplifying the equation, we find:
d ≈ 0.0178 m
Therefore, the magnetic field of 2.4 mT occur at a distance of approximately 0.0178 meters from the wire.