How do two capacitors share the same current?

Question

I am first trying to find the voltages across two capacitors in this circuit.

So first I found i using current division and got 2mA. But when I went to calculate the two separate capacitor voltages, the equations were

v1 = 2000i and v2 = 4000i

In the image on the left I drew the arrows indicating were the currents would split and be different. How can you calculate the voltage across the capacitors using current i which doesn't travel through either branch?

Answer 1

It just happens to work out that way.

Since there's no current through the 5 kohm resistor, you know the voltage \$v_2\$ is equal to the voltage across the 4 kohm resistor, which is of course \$(4\ {\rm k\Omega})i\$.

Also since there's no current through the 5 kohm resistor, you know the current through the 2 kohm resistor is the same as \$i\$, and \$v_1\$ is the same as the voltage across it, so that's \$(2\ {\rm k\Omega})i\$.