Electric Fields And Gauss Law

Consider a dipole consisting of two equal and opposite charges and , separated by a distance . The dipole moment is defined as: The electric field due to a dipole at any point is derived by considering the contribution from both charges. Electric Field on the Equatorial Plane: On the equatorial plane of the dipole, the angle between the position vector and the dipole moment is , and the distance from the dipole is . 1. The expression for the electric field at a point on the equatorial plane at a distance from the center of the dipole is: where is the dipole moment, is the distance from the center of the dipole, and is the permittivity of free space. 2. Direction of the Electric Field: The electric field on the equatorial plane is directed perpendicular to the axis of the dipole and lies in the plane containing the dipole charges. Specifically, it points away from the dipole axis. (I) Electric Field at the Centre of the Dipole ( ): At the center of the dipole, the electric field due to each charge is equal in magnitude but opposite in direction. Therefore, the net electric field at the center of the dipole is zero. Thus, the electric field at the center of the dipole is: (II) Electric Field at a Point : When the distance is much greater than the separation of the charges , the dipole behaves as though it were a point charge. In this case, the electric field behaves as: For large distances, the dipole field behaves like the field due to a point charge with the same total charge . However, for , the field expression becomes much weaker (as ) compared to that of a single charge. Hence, the electric field at a large distance from the dipole is: Thus, at points where , the dipole field decreases rapidly with the cube of the distance.

We are given an electric field N/C, where is the position along the x-axis. The flux through the cube can be calculated using Gauss's law, which states that the net electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space . Mathematically: where is the vector area element on the surface of the cube. Step 1: Electric Field Expression The electric field is given as . This means the electric field has a component along the x-axis, and it depends on the x-coordinate. Step 2: Flux Through Each Face The cube has six faces, and the flux through each face depends on the electric field and the orientation of the face. The area vector for each face is perpendicular to the surface, and the flux through each face is calculated by the dot product . For a face of the cube, the electric flux is given by: where is the area of the face of the cube. Since the electric field is only along the x-axis, the flux through the faces that are parallel to the yz-plane (i.e., the faces at and ) will contribute to the total flux. Step 3: Calculate the Flux Through the Faces at and 1. Face at : The electric field at is N/C. The area of the face at is , and the area vector is in the negative x-direction. Therefore, the flux through this face is: 2. Face at : The electric field at is N/C. The area vector is in the positive x-direction, so the flux through this face is: Step 4: Total Flux The total flux through the cube is the sum of the flux through the two faces (at and ): Thus, the net electric flux through the cube is .