A dipole kept at the origin makes an angle alpha with the z axis as shown in figure. Calculate the electric flux through the upper hemisphere of a spherical surface of radius R centre’d at the origin

To calculate the electric flux through the upper hemisphere of a spherical surface centered at the origin, where a dipole is positioned at the origin making an angle α with the z-axis, we can use Gauss's law and the properties of electric fields generated by dipoles. Let's break this down step by step.

Understanding the Dipole's Electric Field

A dipole consists of two equal and opposite charges separated by a small distance. The electric field \(\mathbf{E}\) due to a dipole at a point in space can be expressed as:

\(\mathbf{E} = \frac{1}{4\pi\epsilon_0} \cdot \frac{2\mathbf{p} \cdot \hat{r} - \mathbf{p}}{r^3}\)

where \(\mathbf{p}\) is the dipole moment vector, \(\hat{r}\) is the unit vector in the direction from the dipole to the point of interest, and \(r\) is the distance from the dipole to that point.

Setting Up the Problem

In our scenario, the dipole moment \(\mathbf{p}\) makes an angle α with the z-axis. The dipole moment can be expressed in Cartesian coordinates as:

\(\mathbf{p} = p \sin(\alpha) \hat{x} + p \cos(\alpha) \hat{z}\)

We want to find the electric flux through the upper hemisphere of a sphere of radius \(R\). The electric flux \(\Phi_E\) through a surface is given by:

\(\Phi_E = \iint_S \mathbf{E} \cdot d\mathbf{A}\)

Calculating the Electric Flux

For a dipole, the electric field is not uniform, and we need to consider the geometry of the upper hemisphere. The area element \(d\mathbf{A}\) on the surface of the hemisphere points outward, which means it has a component in the z-direction. The area element in spherical coordinates can be expressed as:

\(d\mathbf{A} = R^2 \sin(\theta) d\theta d\phi \hat{r}\)

Here, \(\theta\) is the polar angle (from the z-axis) and \(\phi\) is the azimuthal angle. For the upper hemisphere, \(\theta\) ranges from 0 to \(\frac{\pi}{2}\) and \(\phi\) from 0 to \(2\pi\).

Integrating the Electric Field

To find the flux, we need to integrate the dot product of the electric field and the area element over the hemisphere:

\(\Phi_E = \int_0^{2\pi} \int_0^{\frac{\pi}{2}} \mathbf{E} \cdot d\mathbf{A}\)

Substituting the expressions for \(\mathbf{E}\) and \(d\mathbf{A}\), we can simplify the integral. The symmetry of the dipole field and the spherical surface will help us evaluate this integral more easily.

Using Symmetry

Due to the symmetry of the dipole field, the contributions to the flux from the azimuthal angles will cancel out, and we only need to consider the component of the electric field in the z-direction. The z-component of the electric field can be derived from the dipole field equations, leading to:

\(E_z = \frac{1}{4\pi\epsilon_0} \cdot \frac{p \cos(\theta - \alpha)}{R^3}\)

Now, we can compute the flux through the upper hemisphere by integrating this component over the specified limits.

Final Expression for Electric Flux

After performing the integration, the electric flux through the upper hemisphere can be expressed as:

\(\Phi_E = \frac{p}{4\pi\epsilon_0 R^2} \cdot \left(1 - \cos(\alpha)\right)\)

This result shows how the orientation of the dipole affects the electric flux through the hemisphere. The angle α plays a crucial role in determining the amount of flux that passes through the surface.

In summary, by understanding the electric field of a dipole and applying the principles of electric flux, we can derive a clear expression for the flux through a spherical surface. This approach not only highlights the importance of geometry in electromagnetism but also reinforces the concept of electric fields generated by dipoles.