A positive charge +q is placed at the centre of a hollow metallic sphere of inner radius a and outer radius b. the electric field at a distance r from the centre is denoted by E. In this regards, which one of the following statement is correct?
E = 0 for a < r < b
Understanding Electric Field in a Hollow Metallic Sphere
This question asks about the electric field at different distances from the center of a hollow metallic sphere. A positive charge +q is placed exactly at the center. The sphere has an inner radius 'a' and an outer radius 'b'. We need to analyze the electric field E at a distance 'r' from the center.
Electric Field in Different Regions
Let's consider the electric field in three different regions based on the distance 'r' from the center:
- Region 1: r < a (Inside the cavity)
- Region 2: a < r < b (Inside the metallic material)
- Region 3: r > b (Outside the sphere)
Region 1: Electric Field for r < a (Inside the cavity)
In this region, we are inside the hollow cavity but outside the central charge. Since the charge +q is at the center, we can use Gauss's Law. Consider a spherical Gaussian surface of radius 'r' where r < a. The charge enclosed by this surface is +q.
According to Gauss's Law, the electric flux through a closed surface is equal to the enclosed charge divided by $\varepsilon_0$:
$\oint \vec{E} \cdot d\vec{A} = \frac{q_{enclosed}}{\varepsilon_0}$
Due to spherical symmetry, the electric field $\vec{E}$ is radial and has the same magnitude E at all points on the Gaussian surface. The area vector $d\vec{A}$ is also radial. So, $\vec{E} \cdot d\vec{A} = E \, dA$.
$E \oint dA = \frac{q}{\varepsilon_0}$
$E (4\pi r^2) = \frac{q}{\varepsilon_0}$
$E = \frac{q}{4\pi\varepsilon_0 r^2}$ for $r < a$ (but $r \neq 0$).
The electric field is non-zero in this region and decreases with the square of the distance.
Region 2: Electric Field for a < r < b (Inside the metallic material)
This region is inside the metallic material of the sphere. A key property of conductors in electrostatic equilibrium is that the electric field inside the conductor is always zero. This is because free charges within the conductor rearrange themselves in response to any external electric field until the net field inside is zero.
When the charge +q is placed at the center, it induces a charge -q on the inner surface (at r=a) and a charge +q on the outer surface (at r=b) of the metallic sphere. The induced charges distribute themselves such that they cancel out the field due to the central charge everywhere within the metallic material.
Therefore, the electric field E is zero for a < r < b.
Region 3: Electric Field for r > b (Outside the sphere)
Consider a spherical Gaussian surface of radius 'r' where r > b. The total charge enclosed by this surface is the sum of the charge at the center (+q) and the induced charges on the inner (-q) and outer (+q) surfaces of the sphere.
$q_{enclosed} = (+q_{center}) + (-q_{inner}) + (+q_{outer}) = +q + (-q) + (+q) = +q$
Using Gauss's Law as before:
$E (4\pi r^2) = \frac{q_{enclosed}}{\varepsilon_0} = \frac{q}{\varepsilon_0}$
$E = \frac{q}{4\pi\varepsilon_0 r^2}$ for $r > b$.
The electric field outside the sphere is the same as if the total charge +q were concentrated at the center.
Analyzing the Given Statements about Electric Field E
Now let's look at the provided statements and compare them with our findings:
- Statement 1: E = 0 for a < r < b
This statement says the electric field is zero inside the metallic sphere. As explained above, the electric field inside a conductor in electrostatic equilibrium is indeed zero. This statement is correct. - Statement 2: E = 0 for r < a
This statement says the electric field is zero inside the cavity. For r < a (but r $\neq$ 0), we found that $E = \frac{q}{4\pi\varepsilon_0 r^2}$, which is not zero. This statement is incorrect. - Statement 3: E = q / 4πϵ 0r for a < r < b
This statement provides a non-zero formula for the electric field inside the metallic sphere. As we know, E = 0 inside the conductor (a < r < b). Also, the formula provided $E \propto 1/r$ is incorrect dimensionally for the magnitude of the electric field from a point charge which varies as $1/r^2$. This statement is incorrect. - Statement 4: E = q / 4πϵ 0r for r < a
This statement provides a formula for the electric field inside the cavity. We found that $E = \frac{q}{4\pi\varepsilon_0 r^2}$ for r < a. The formula provided $E \propto 1/r$ is incorrect dimensionally. While the field is non-zero, this specific formula is wrong. This statement is incorrect.
Based on the analysis, only the first statement correctly describes the electric field in the specified region.
Conclusion on the Correct Statement
The correct statement regarding the electric field E at a distance r from the center of the hollow metallic sphere with a positive charge +q at its center is that E = 0 for a < r < b. This is a fundamental property of conductors in electrostatic equilibrium.
| Region | Distance r | Electric Field Magnitude E |
|---|---|---|
| Inside the cavity | r < a | $\frac{q}{4\pi\varepsilon_0 r^2}$ (for r $\neq$ 0) |
| Inside the metal | a < r < b | 0 |
| Outside the sphere | r > b | $\frac{q}{4\pi\varepsilon_0 r^2}$ |
Revision Table: Electric Field in Spherical Systems
| Scenario | Location | Electric Field E | Key Concept |
|---|---|---|---|
| Point charge q | distance r | $\frac{q}{4\pi\varepsilon_0 r^2}$ | Coulomb's Law, Gauss's Law |
| Charged spherical shell (radius R, charge Q) | r < R | 0 | Gauss's Law (enclosed charge is 0) |
| Charged spherical shell (radius R, charge Q) | r > R | $\frac{Q}{4\pi\varepsilon_0 r^2}$ | Gauss's Law (enclosed charge is Q) |
| Solid conducting sphere (radius R, charge Q) | r < R | 0 | Electrostatic equilibrium in conductor |
| Solid conducting sphere (radius R, charge Q) | r > R | $\frac{Q}{4\pi\varepsilon_0 r^2}$ | Gauss's Law (enclosed charge is Q) |
| Hollow metallic sphere with charge at center (as in question) | a < r < b (inside metal) | 0 | Electrostatic equilibrium in conductor |
Additional Information on Conductors and Electrostatic Shielding
This problem highlights the properties of conductors in electrostatic equilibrium. When a conductor is in electrostatic equilibrium (no net motion of charges), several important conditions are met:
- The electric field is zero everywhere inside the conductor.
- Any net charge on a conductor resides entirely on its surface(s).
- The electric potential is constant throughout the volume of the conductor and is equal to the potential on its surface.
- Electric field lines are perpendicular to the surface of the conductor.
In our case, the positive charge +q at the center induces charges on the inner and outer surfaces of the hollow metallic sphere. A charge -q is induced on the inner surface (r=a) facing the central charge. This is because the positive charge attracts free electrons in the conductor to the inner surface. Since the conductor was initially neutral (implicitly, as no net charge was mentioned for the sphere itself), a charge +q is left behind on the outer surface (r=b) to maintain overall neutrality of the sphere.
The phenomenon where the electric field is zero inside a hollow conductor cavity, even if there are external fields, is known as electrostatic shielding. In this specific problem, the field in the cavity is due to the charge *inside* it, but the metallic shell itself ensures the field is zero within the metal material.
Important Questions from Electric Fields and Gauss' Law
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