Question #43354

Three concentric thin conducting spherical shells of radii $a$, $2a$, and $3a$ are arranged in a vacuum. The innermost shell is grounded. The middle shell is given a charge $+Q$ and the outermost shell is given a charge $-Q$. The potential difference between the middle and outermost shells is connected across a secondary circuit to find its null point on a potentiometer. The potentiometer consists of a uniform wire of length $L$ and resistance $R$, driven by a primary ideal battery of EMF $E_0$ and a series resistance of $R/2$. If the null point is obtained at a distance $x$ from the high-potential end of the potentiometer wire, what is the value of $x$? (Assume the secondary circuit draws no current at the null point and $x < L$).