= Dielectric insertion force at fixed voltage
{title2=$F_x=V^2C'(x)/2$}
At fixed <electric potential difference> $V$, a change of <capacitance> has electrical work input $V\,dQ=V^2dC$ and field-energy increase $dU=V^2dC/2$. The remaining half is mechanical <work> done by the field, so the force in the direction of increasing overlap is $F_x=V^2C'(x)/2$. Equivalently the effective mechanical potential is $-CV^2/2$. For a <partially filled coaxial capacitor>, this gives $F_x=\pi\epsilon_0(\epsilon_r-1)V^2/\ln(b/a)$, directed inward when $\epsilon_r>1$. The <fringing field of a capacitor> supplies the local field gradient and transmits this force even when its small correction to total capacitance is neglected.
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