Confusion in electric field inside a conductor

Question

Image I'm referring to:

In the figure both shells are conductors.

Let's say I have to find the electric field at a point P as shown in the figure. I thought, since this point lies inside a conductor, the Electric field at this point has to be $0$ no matter what. But I was taught that there will be no electric field at P due to the outer shell but there will be a electric field at point P due to the inner shell, (because it has some charge).

I'm confused now, beacuse earlier I was told the electric field inside a conductor is $0$, and that point clearly lies inside the conductor, where am I going wrong? Won't there be induction of $-Q_{1}$ on the inner surface of the outer shell, in such a way that the electric field inside the conductor becomes $0$?

Also, by the term inside the conductor mean in the bulk or in the material of the conductor or something else?

Please help me out, I have studied from many sources, any no one cares to be critical about their words, hence the confusion.

Edit: For all future readers, this image is a good explaination of what is going on inside such situations.

Answer 1

The most reliable way to approach these problems is to use fundamental laws of electromagnetism, in particular Gauss' law and linearity of the electric field. A thorough application of these laws can answer all of your questions.

Gauss law

The electric field flux through a closed surface $\Phi_E$ is given by the charge inside that surface $Q_{int}$ divided by $\varepsilon_0$: $$\Phi_E = \frac{Q_{int}}{\varepsilon_0 }. $$

Linearity of electric field

The total electric field at a point is the vector sum of all the electric field produced by the single sources. If there are two sources labeled $1$ and $2$, the field at a point $\vec{E}(P)$ is

$$ \vec{E}(P) = \vec{E}_1(P) + \vec{E}_2(P). $$

Let's now use these tools to solve your problem: at point P there are in principle two contributions to the field, one coming from the inner shell, and one coming from the outer shell. The former has modulus $Q_1/4\pi\varepsilon_0r^2$ and the latter vanishes, as a consequence of Gauss' law (exercise). Bottom line: there is an electric field at P, only coming from the inner shell charges.

Now let's comment your statements one by one:

I thought, since this point lies inside a conductor, the Electric field at this point has to be 0 no matter what.

This is a misunderstanding, since "inside a conductor" means "in the bulk" of a conductor, but here there's no bulk at all.

I was taught that there will be no electric field at P due to the outer shell but there will be a electric field at point P due to the inner shell

This is correct, as discussed above.

EDIT while writing this answer, I was unaware of most of the comments and other answers, so there might be some redundancy.