Capacitor with a Dielectric — Question 9

Question 9

Thermal expansion can cancel the first-order dielectric temperature coefficient

A plate capacitor is filled with a dielectric obeying $\kappa(T)=\kappa_0(1+\beta\Delta T)$. Every geometric length scales by $1+\alpha\Delta T$. Treat these supplied laws as exact over the small interval and use $C=\kappa\epsilon_0A/d$.

1. Find the exact capacitance ratio and its first-order temperature coefficient.

2. Choose beta to cancel first-order drift and find the remaining fractional change.

Question 9 - Solution

Thermal expansion can cancel the first-order dielectric temperature coefficient

(a) Method and calculation

$$\frac{C(T)}{C(0)}=(1+\beta\Delta T)\frac{(1+\alpha\Delta T)^2}{1+\alpha\Delta T}=(1+\beta\Delta T)(1+\alpha\Delta T).$$$$\frac{\Delta C}{C(0)}=(\alpha+\beta)\Delta T+\alpha\beta(\Delta T)^2.$$

(b) Interpretation and further calculation

$$\beta=-\alpha\Rightarrow\left.\frac1C\frac{dC}{dT}\right|_{\Delta T=0}=0.$$$$\frac{\Delta C}{C(0)}=-\alpha^2(\Delta T)^2.$$

Physical check

Cancellation of the linear drift does not remove the quadratic term in the explicitly supplied material and expansion model.