Field inside a cavity of a conductor Consider a cavity in any randomly shaped Conductor as shown in the figure

Where Eo is the field due to σo
And, Ei is the field due to σi
My Question is,
Why don't we take any contribution of Ei when calculating field at any point inside the cavity ?
i.e
Why is the Total field at a point inside the cavity is given by only Ec ?
 A: It seems like you're assuming that the electric field inside the cavity, $\vec{E}_c$, is due only to the charge inside the cavity (let's call it $\rho_c$.)    This is false.  The easiest way to see this is by looking at an example where a conducting cavity has an "off-center" charge in it:

(diagram cribbed from this PhysicsForums thread)
The electric field $\vec{E}_c$ due to the point charge inside the cavity is just a radial field.  But the electric field must be normal to a conducting surface when we're just outside that surface;  and the field from an off-center charge inside a spherical cavity won't have this property.  So it can't be the case that the electric field inside the cavity is due only to the charges inside the cavity.  Instead, the electric field inside the cavity is due to both the point charge in the center and the induced charges on the inner surface:
$$\vec{E} = \vec{E}_c + \vec{E}_i$$
Really, one should think of the electric field inside the cavity as including the electric fields due to the outer surface charge and the rest of the Universe as well:
$$
\vec{E} = \vec{E}_c + \vec{E}_i + \vec{E}_o + \vec{E}_u
$$
But because of the properties of conductors, $\vec{E}_o$ and $\vec{E}_u$ cancel out exactly;  so we can get away with only considering $\vec{E}_c$ and $\vec{E}_i$ to calculate the field inside the cavity.
