Given

The charge density in the region between two concentric spheres of radii $a$ and $b$ is $\rho(r)=\frac{A}{r}$ where $a<r<b$.

We need to find $A$ such that the electric field in the region between the spheres is constant.

Use Gauss’s law

For a spherical Gaussian surface of radius $r$ such that $a<r<b$, $E(4\pi r^2)=\frac{Q_{\text{enc}}}{\varepsilon_0}$ so $E=\frac{Q_{\text{enc}}}{4\pi\varepsilon_0 r^2}.$

Find enclosed charge up to radius $r$

Since the charge exists only from $a$ to $r$, $Q_{\text{enc}}=\int_a^r \rho(r')\, dV$ with $dV=4\pi r'^2 dr'.$

Therefore, $Q_{\text{enc}}=\int_a^r \frac{A}{r'}\cdot 4\pi r'^2\,dr'$ $=4\pi A\int_a^r r'\,dr'$ $=4\pi A\left[\frac{r'^2}{2}\right]_a^r$ $=2\pi A(r^2-a^2).$

Substitute into electric field expression

$E=\frac{2\pi A(r^2-a^2)}{4\pi\varepsilon_0 r^2}$ $E=\frac{A}{2\varepsilon_0}\left(1-\frac{a^2}{r^2}\right).$

This is not constant for general $r$ because of the term $\dfrac{a^2}{r^2}$.

Interpretation of the question

For the electric field in the shell region to be constant, the usual intended condition in such problems is that the total enclosed charge up to radius $r$ should be proportional to $r^2$.

But with charge distributed only in the shell $a<r<b$, we instead use the total charge in the shell to determine $A$ corresponding to the standard result used in such JEE problems.

Let total charge in the shell be $Q$.

Then $Q=\int_a^b \frac{A}{r}\,4\pi r^2dr$ $=4\pi A\int_a^b r\,dr$ $=4\pi A\left[\frac{r^2}{2}\right]_a^b$ $=2\pi A(b^2-a^2).$

Hence, $A=\frac{Q}{2\pi(b^2-a^2)}.$

Match with options

This corresponds to $\boxed{\frac{Q}{2\pi(b^2-a^2)}}$ which is Option D.

Compare with stored answer

Stored correct answer is C: $\frac{Q}{2\pi a^2}$

But from direct integration of the given charge density over the shell, the correct value is $\boxed{\frac{Q}{2\pi(b^2-a^2)}}$ so the stored answer does not match the derivation.

Note on consistency

Also, strictly speaking, with charge density $\rho=A/r$ only in the region $a<r<b$, the electric field inside the shell is $E=\frac{A}{2\varepsilon_0}\left(1-\frac{a^2}{r^2}\right),$ which is not constant. So the wording of the problem seems inconsistent. However, among the given options, the physically derived value of $A$ in terms of total shell charge $Q$ is Option D.