B

According to the capillary rise formula, the height of liquid risen in a capillary tube is given by the formula:
$$ h = \frac{2\gamma cos(\theta)}{\rho g r} $$
where:
- $h$ is the height of the liquid rise,
- $\gamma$ is the surface tension of the liquid,
- $\theta$ is the angle of contact,
- $\rho$ is the density of the liquid,
- $g$ is the acceleration due to gravity,
- $r$ is the radius of the capillary tube.
When the diameter is doubled, the radius $r$ also doubles (since $r = \frac{d}{2}$).
If the new radius is $r' = 2r$, we can substitute this into the formula for height:
$$ h' = \frac{2\gamma cos(\theta)}{\rho g r'} = \frac{2\gamma cos(\theta)}{\rho g (2r)} = \frac{1}{2} \left( \frac{2\gamma cos(\theta)}{\rho g r} \right) = \frac{1}{2} h $$
Therefore, when the diameter is doubled, the height of the liquid that will rise is half of the original height.
Therefore, the correct answer is B.