Two wires A and B of same length and of the same material have the respective radii r₁ and r₂. Their one end is fixed with a rigid support, and at the other end equal twisting couple is applied. Then the ratio of the angle of twist at the end of A and the angle of twist at the end of B will be

To find the ratio of the angle of twist at the end of wire A to the angle of twist at the end of wire B, we use the relation for the twisting couple (torque) applied to a cylindrical wire.

The twisting couple (torque) $\tau$ required to twist a solid cylinder of length $l$, radius $r$, and material shear modulus $\eta$ through an angle $\theta$ is given by the formula:
$\tau =\frac{\pi \eta r^4\theta }{2l}$
where:
• $\tau$ is the applied twisting couple,
• $\eta$ is the modulus of rigidity of the material,
• $r$ is the radius of the wire,
• $\theta$ is the angle of twist, and
• $l$ is the length of the wire.

Rearranging the equation to solve for the angle of twist $\theta$:
$\theta =\frac{2l\tau }{\pi \eta r^4}$

According to the problem description:
1. Both wires have the same length: $l_1=l_2=l$
2. Both wires are made of the same material, which means they have the same modulus of rigidity: ${\eta }_1={\eta }_2=\eta$
3. Equal twisting couples are applied: ${\tau }_1={\tau }_2=\tau$

Since $l$, $\eta$, and $\tau$ are constant for both wires, the angle of twist $\theta$ is inversely proportional to the fourth power of the radius $r$:
$\theta \propto \frac{1}{r^4}$

Let ${\theta }_1$ and ${\theta }_2$ be the angles of twist for wires A and B respectively, and let $r_1$ and $r_2$ be their respective radii. The ratio of the angle of twist of A to that of B is:
$\frac{{\theta }_1}{{\theta }_2}=\frac{r_2^4}{r_1^4}$

Therefore, the ratio of the angle of twist at the end of A and the angle of twist at the end of B is $r_2^4/r_1^4$.