Energy Changes in Capacitor After Disconnecting from Battery

[MatinSAR]

Homework Statement

A parallel-plate capacitor with a plate separation of $d$ is connected to a battery, having energy $U_1$. A dielectric with a constant $k=2$ is inserted between the capacitor's plates, and the plate separation is reduced by 20%. The capacitor is then disconnected from the battery, and the dielectric is removed. The energy stored in the capacitor changes to $U_2$. What is the ratio $U_1/U_2$?

Relevant Equations

$q = CV$

My try:

At first, the energy is $U_1 = \dfrac {q^2_1}{2C_1}$. After inserting the dielectric and reducing the distance between plates, the capacitance changes to $\dfrac {5}{2}C_1$, and because the voltage is constant, we have $q_2 = \dfrac {5}{2}q_1$. When we disconnect it from the battery and remove the dielectric, the charge remains unchanged but both capacitance and voltage start changing... So $q_2 = \dfrac {5}{2}q_1$ and $C_2 = \dfrac {5}{4}C_1$, so $U_2 = \dfrac {5}{2}U_1$, and therefore $U_1/U_2 = \dfrac {2}{5} = 0.4$.

But the options are:

1. 25/4
2. 25/16
3. 4/25
4. 16/25

 

[Steve4Physics]

... So $q_2 = \dfrac {5}{2}q_1$ and $C_2 = \dfrac {5}{4}C_1$

Agreed.

so $U_2 = \dfrac {5}{2}U_1$, and therefore $U_1/U_2 = \dfrac {2}{5} = 0.4$.

Check - remember $U = \frac 12 \frac {Q^2}C$.

But, having said that, I don't get any of the answers in the list.

 

[MatinSAR]

Check - remember $U = \frac 12 \frac {Q^2}C$.

Yes I forget that $U_1$ has $2$ in denominator.
$$U_2 = \dfrac {(25/4)q^2_1}{(5/2)C_1}= (5/2)(q^2_1/C_1)$$$$U_1 = \dfrac {q^2_1}{2C_1}$$$$U_1/U_2 = 1/5$$
Do you agree with my reasoning?

 

[Steve4Physics]

Yes I forget that $U_1$ has $2$ in denominator.
$$U_2 = \dfrac {(25/4)q^2_1}{(5/2)C_1}= (5/2)(q^2_1/C_1)$$$$U_1 = \dfrac {q^2_1}{2C_1}$$$$U_1/U_2 = 1/5$$
Do you agree with my reasoning?

Yes. FWIW I like to use proportionality for this type of problem. Down to personal preferences of course.

$U = \frac {Q^2}{2C}$. Since $Q$ changes by a factor $\frac 52$ and $C$ changes by a factor $\frac 54$, $U$ changes by a factor $\frac {{(\frac 52)^2}}{{\frac 54}} = 5$. I.e. $U_2 = 5U_1$.

 

[MatinSAR]

@Steve4Physics Thank you for your help.