The mag. field (B) at the centre of a circular coil of radius "a", through which a current I flows is (a) \(\mathrm{B} \propto \mathrm{c} \mathrm{a}\) (b) \(\mathrm{B} \propto(1 / \mathrm{I})\) (c) \(\mathrm{B} \propto \mathrm{I}\) (d) \(B \propto I^{2}\)

The Biot-Savart Law gives the magnetic field produced by a small current element, and is given by: \[ d\vec{B} = \frac{\mu_0}{4\pi} \cdot \frac{I \cdot d\vec{l} \times \vec{r}}{r^3} \] where \(d\vec{B}\) is the magnetic field produced by the current element, \(\mu_0\) is the permeability of free space, I is the current flowing through the current element, \(d\vec{l}\) is the length of the current element, \(\vec{r}\) is the position vector from the current element to the point where we want to find the magnetic field, and r is the magnitude of \(\vec{r}\).

The given coil is a closed circular loop, so we need to integrate the Biot-Savart Law along the entire loop to find the total magnetic field at the center. Let's consider the center of the loop to be at the origin (0, 0, 0). The position vector \(\vec{r}\) will have the same magnitude "a" for all points on the loop, and the angle between \(d\vec{l}\) and \(\vec{r}\) at each point will be 90 degrees. So, the cross product simplifies to: \[ d\vec{l} \times \vec{r} = |d\vec{l}| |\vec{r}| \sin{90^{\circ}} = adl \] Now we need to integrate around the entire loop: \[ B = \oint \frac{\mu_0}{4\pi} \cdot \frac{I \cdot adl}{a^3} = \frac{\mu_0}{4\pi} \cdot \frac{I \cdot a}{a^3} \oint dl \] Since the integral of the length element \(dl\) around the whole loop is simply the circumference of the loop (2πa), we have: \[ B = \frac{\mu_0}{4\pi} \cdot \frac{I \cdot a}{a^3} \cdot 2\pi a = \frac{\mu_0I}{2a} \]

Now we have found the magnetic field B at the center of the loop: \[ B = \frac{\mu_0I}{2a} \] From this result, we can find the correct proportionality relation among the given options. (a) B ∝ ca: This option implies that B is proportional to the radius a, but here B is inversely proportional to a. (b) B ∝ (1/I): This option implies that B is inversely proportional to I, but here B is directly proportional to I. (c) B ∝ I: This matches our result since B is directly proportional to I. (d) B ∝ I²: This option implies that B is proportional to the square of I, but our result shows it's directly proportional to I, not its square. Therefore, the correct answer is (c) B ∝ I.