How large a current would a very long, straight wire have to carry so that the magnetic field 2 cm from the wire is equal to 1 G (comparable to the earth’s northward-pointing magnetic field)?
To determine the required current, you can use Ampere's Law, which states that the magnetic field produced by a long, straight wire is directly proportional to the current flowing through it.
The formula for calculating the magnetic field around a long, straight wire is given by:
B = (μ0 * I) / (2 * π * r)
Where:
B is the magnetic field,
μ0 is the permeability of free space (4π * 10^-7 T*m/A),
I is the current,
r is the distance from the wire.
Given that the magnetic field (B) is 1 G (or 1 × 10^-4 T) and the distance (r) from the wire is 2 cm (or 0.02 m), we can rearrange the formula to solve for the current (I):
I = (B * 2 * π * r) / μ0
Substituting the given values:
I = (1 × 10^-4 T * 2 * π * 0.02 m) / (4π * 10^-7 T*m/A)
Simplifying:
I = (1 × 10^-6 T*m) / (4π * 10^-7 T*m/A)
Dividing the numerator and denominator by 10^-6:
I = (1 * 10) / (4π)
Simplifying further and evaluating:
I ≈ 0.0796 A
Therefore, a very long, straight wire would have to carry approximately 0.0796 A of current to produce a magnetic field of 1 G at a distance of 2 cm from the wire.
To determine the current required for a specific magnetic field at a certain distance from a wire, we can use Ampere's Law. Ampere's Law states that the magnetic field around a long, straight wire is proportional to the current passing through the wire and inversely proportional to the distance from the wire.
The formula to calculate the magnetic field generated by a long, straight wire is:
B = (μ0 * I) / (2π * r)
Where:
B is the magnetic field strength (in Tesla)
μ0 is the permeability of free space, approximately 4π x 10^-7 T m / A
I is the current in the wire (in Amperes)
r is the distance from the wire (in meters)
In this case, we want to find the current (I) when the magnetic field (B) is equal to 1 G (0.0001 T) at a distance of 2 cm (0.02 m) from the wire.
Substituting the given values into the formula, we have:
0.0001 T = (4π x 10^-7 T m / A) * I / (2π * 0.02 m)
Simplifying the equation:
0.0001 T = (2 x 10^-7 T m / A) * I
Canceling out the units and rearranging the equation:
I = (0.0001 T) / (2 x 10^-7 T m / A)
Calculating the current:
I = 0.5 A
Therefore, a very long, straight wire would need to carry a current of 0.5 Amperes to generate a magnetic field of 1 Gauss (0.0001 Tesla) at a distance of 2 cm from the wire.