# What is the mistake in this derivation of capillary rise, using equal pressures at equal height?

I have seen derivations of capillary height using force equations:

$$2\pi RT \cos{\theta} = \rho \pi R^{2}hg$$

which gives,

$$h = \frac{2T\cos{\theta}}{\rho gR}$$

Now, if we go about this another way, the pressure at the free surface $$P_{\text{atm}}$$ should be equal to the pressure inside the capillary tube, at the same level.

Or,

$$P_{\text{atm}} - \frac{2T}{R} + ρgh = P_{\text{atm}}$$

This will give:

$$h = \frac{2T}{\rho gR},$$ which is wrong.

What is the mistake in this alternate derivation?