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 variation of magnetic field strength from the wire surface to a distance of 40mm from the centre of the wire, we need to understand the relationship between the magnetic field strength and the distance from the wire.

The magnetic field strength around a long straight wire carrying a steady current can be calculated using Ampere's law. Ampere's law states that the magnetic field strength at a distance r from an infinitely long straight wire carrying a current I 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 Tm/A), I is the current, and r is the distance from the wire.

Now, let's calculate the magnetic field strength at the surface of the wire (a) and at a distance of 40mm from the center of the wire (b).

(a) The surface of the wire is at a distance of 0mm from the center. Plugging this value into the formula:

B = (μ₀ * I) / (2π * r)

We can calculate B as:

B = (4π x 10^-7 Tm/A * I) / (2π * 0mm)

We need to remember that 1mm = 0.001m.

Since the diameter of the wire is given as 2.0mm, the radius of the wire is 1.0mm (0.001m).

Plugging in the numbers and solving the equation:

B = (4 * 3.14 x 10^-7 Tm/A * I) / (2 * 3.14 * 0.001m)

B = (2 x 10^-7 T * m/A * I) / (0.002m)

B = (10^-7 T * m/A * I) / (0.001m)

So, the magnetic field strength at the surface of the wire is:

B = (10^-7 T * m/A * I) / (0.001m)

Now, let's move on to the next part.

(b) The distance from the center of the wire is given as 40mm = 0.04m. Plugging this value into the formula:

B = (μ₀ * I) / (2π * 0.04m)

Again, plugging in values and solving the equation:

B = (4 * 3.14 x 10^-7 Tm/A * I) / (2 * 3.14 * 0.04m)

B = (2 x 10^-7 T * m/A * I) / (0.08m)

B = (10^-7 T * m/A * I) / (0.04m)

Thus, the magnetic field strength at a distance of 40mm from the center of the wire is:

B = (10^-7 T * m/A * I) / (0.04m)

I hope this helps you understand how to calculate and sketch the variation of magnetic field strength from a wire surface to a distance of 40mm from the center of the wire.