A long straight vertical wire of diameter 2.0mm carries a steady current upwards. At 30mm from the centre of the wire, the magnetic field strength is 0.10mT. Sketch a graph to show the variation of magnetic field strength from the wire surface to a distance of 40mm from the centre of the wire. Calculate the magnetic field strength at

(a) the surface of the wire,

(b) 40mm from the centre of the wire.

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.