Of the four graphs provided, which one best represents the...

The relation between the current $i$ and the angle of deflection $\theta$ in a tangent galvanometer is given by:

$i = K \tan \theta$

where $K$ is the reduction factor of the galvanometer. Rearranging the formula for $\theta$, we get:

$\theta = \tan^{-1} \left( \dfrac{i}{K} \right)$

Analyzing the function $\theta(i)$:
1. At $i = 0$, $\theta = \tan^{-1}(0) = 0$. The graph passes through the origin.
2. For small values of $i$, $\theta \approx \dfrac{i}{K}$, so the graph initially has a constant positive slope.
3. As $i \to \infty$, $\theta \to \dfrac{\pi}{2}$. The graph has a horizontal asymptote at $\theta = \dfrac{\pi}{2}$.
4. The slope is $\dfrac{d\theta}{di} = \dfrac{K}{K^2 + i^2}$, which decreases as $i$ increases. This means the curve is concave downward.

Observing the given graphs, curve 'c' starts from the origin, is concave downward, and asymptotically approaches the value $\dfrac{\pi}{2}$ on the vertical axis.

Thus, graph 'c' best represents the current-deflection relation.