To find the intensity of the electric field at the center of a hollow dielectric cylinder that is uniformly polarized, we need to analyze the situation step by step. The cylinder has an inner radius \( a \), an outer radius \( b \), and a height of \( 2h \). The polarization vector \( \mathbf{P} \) is directed normally to the bases of the cylinder, and the surrounding medium is vacuum. Let's break down the problem and derive the expression for the electric field at point C, which is located at the center of the cylinder.
Understanding Polarization and Electric Field
When a dielectric material is polarized, it develops bound charges on its surfaces. In this case, the polarization vector \( \mathbf{P} \) creates a surface charge density on the inner and outer surfaces of the cylinder. The bound surface charge densities can be calculated as follows:
- On the inner surface (radius \( a \)): \( \sigma_{-} = -\mathbf{P} \cdot \hat{n} = -P \) (since the normal is inward).
- On the outer surface (radius \( b \)): \( \sigma_{+} = \mathbf{P} \cdot \hat{n} = P \) (normal is outward).
Calculating the Electric Field
To find the electric field at point C, we can use the principle of superposition. The total electric field at the center will be the vector sum of the fields due to the inner and outer surfaces. The electric field due to a uniformly charged infinite plane sheet is given by:
E = σ / (2ε₀)
For the inner surface at radius \( a \):
The electric field \( E_{inner} \) at point C due to the inner surface is directed towards the center (negative direction), and its magnitude is:
E_{inner} = -\frac{-P}{2ε₀} = \frac{P}{2ε₀}
For the outer surface at radius \( b \):
The electric field \( E_{outer} \) at point C due to the outer surface is directed away from the surface (positive direction), and its magnitude is:
E_{outer} = \frac{P}{2ε₀}
Combining the Electric Fields
Now, we need to consider the contributions of both fields at point C:
E_{total} = E_{inner} + E_{outer}
However, we must account for the distances from point C to the surfaces. The electric field due to a surface charge at a distance \( r \) from the center is modified by the geometry of the cylinder. The effective distances to consider are:
- For the inner surface: \( \sqrt{a^2 + h^2} \)
- For the outer surface: \( \sqrt{b^2 + h^2} \)
Thus, the total electric field at point C becomes:
E_{total} = -\frac{P h}{ε₀} \left( \frac{1}{\sqrt{a^2 + h^2}} - \frac{1}{\sqrt{b^2 + h^2}} \right)
Final Result
In conclusion, the intensity of the electric field at the center of the hollow dielectric cylinder is given by:
E = -\frac{Ph}{ε₀} \left( \frac{1}{\sqrt{a^2 + h^2}} - \frac{1}{\sqrt{b^2 + h^2}} \right)
This formula captures how the electric field at the center is influenced by the polarization of the dielectric and the geometry of the cylinder. Understanding these relationships helps in grasping the behavior of electric fields in polarized materials.
