To sketch the graph of the magnetic field strength, we need to understand how the field strength varies with distance. The magnetic field produced by a long straight wire carrying a steady current follows the inverse-square law.
The formula to calculate the magnetic field strength at a distance from the wire is given by:
B = (µ₀ * I) / (2π * r)
Where:
B is the magnetic field strength
µ₀ is the permeability of free space (4π x 10^-7 T·m/A)
I is the current in the wire
r is the distance from the wire
(a) At the surface of the wire, the distance from the center is the radius of the wire, which is half the diameter. So, r = 1.0 mm (or 0.001 m).
Using the formula, we can calculate the magnetic field strength:
B₁ = (µ₀ * I) / (2π * r)
Substituting the values, we get:
B₁ = (4π x 10^-7 T·m/A * I) / (2π * 0.001 m)
= (2 x 10^-7 T·m/A * I) / (0.001 m)
= 2 x 10^-4 T * I
(b) At a distance of 40 mm from the center of the wire, r = 40 mm = 0.04 m.
Using the formula again:
B₂ = (µ₀ * I) / (2π * r)
Substituting the values, we get:
B₂ = (4π x 10^-7 T·m/A * I) / (2π * 0.04 m)
= (2 x 10^-5 T·m/A * I) / (0.04 m)
= 5 x 10^-4 T * I
To sketch the graph, we can plot the magnetic field strength (B) on the y-axis and the distance from the center of the wire (r) on the x-axis. The graph should show a decrease in magnetic field strength as the distance from the wire increases. The value at the surface of the wire (B₁) can be plotted, as well as the value at 40 mm from the center of the wire (B₂).
Please note that the values I provided are based on the given information. If the current in the wire is provided, you can substitute it into the formulas to calculate the actual magnetic field strengths.