The electrical resistance, R ohms, of a wire of a given length is inversely proportional to the square of the diameter of the wire, d mm. If R = 2.0 ohms when d = 3 mm, find the value of R when d = 4mm.

The problem states that the electrical resistance $R$ is inversely proportional to the square of the diameter $d$. This can be written as:

$$R = \frac{k}{d^2}$$

where $k$ is the constant of proportionality.

We are given that $R = 2.0$ ohms when $d = 3$ mm. Substitute these values into the equation:

$$2.0 = \frac{k}{(3)^2}$$ $$2.0 = \frac{k}{9}$$

Multiply both sides by 9 to solve for $k$:

$$k = 2.0 \times 9$$ $$k = 18$$

Now use the calculated value of $k = 18$ and the new diameter $d = 4$ mm in the relationship:

$$R = \frac{18}{(4)^2}$$ $$R = \frac{18}{16}$$ $$R = 1.125$$

The value of $R$ when $d = 4$ mm is 1.125 ohms.

$$\text{1.125 ohms}$$