Electric Charges And Fields

Electric Field at Point due to a Dipole

Step 1: Distance from Point P to the Charges

Charges are placed at:

• at
• at

Distance from each charge to point is:

Step 2: Electric Field Due to Each Charge

The magnitude of electric field due to a point charge is given by:

Let and be the fields due to and respectively. They make an angle with the vertical, where:

Because of symmetry:

• The horizontal components ( ) cancel each other.
• The vertical components ( ) add up.

Vertical component of the field due to one charge:

Net vertical electric field at point :

Step 3: At a Far-Off Point ( )

If , then . So,

Conclusion:

At large distances along the y-axis, the electric field due to the dipole varies as .

According to Gauss's law, the electric flux through a surface is proportional to the net charge enclosed by the surface: The total charge enclosed by the surface is the sum of the charges inside and the charge placed between and . Therefore, the total charge enclosed by is: The flux through surface is proportional to the charge inside it: The flux through surface is four times the flux through : Using Gauss's law for : Equating the two expressions for : Solving for : Substituting the values of , , and : Thus, the charge is .

Difference Between Open Surface and Closed Surface

1. Definition of Open Surface:

An open surface is a surface that does not enclose a volume. It has at least one boundary or edge and is not completely closed. It is like a sheet or a plane that extends infinitely or ends at a boundary. Examples include:

• A flat sheet of paper
• The surface of water in a glass (which does not form a closed boundary in the 3D space)

An open surface can be used in flux calculations in physics, such as when calculating the electric flux or magnetic flux through a given area.

2. Definition of Closed Surface:

A closed surface is a surface that fully encloses a volume with no boundaries or edges. The surface creates a boundary that completely contains a region of space. It is like the surface of a sphere, cube, or any enclosed 3D object. Examples include:

• The surface of a balloon
• The surface of a sphere or a cube

A closed surface is often used in the Gauss's Law for electric fields or the magnetic flux through a surface, as it encloses a region of space and helps in calculating the net flux through the surface.

3. Key Differences Between Open and Closed Surfaces:

• Boundary: An open surface has a boundary or edge, whereas a closed surface has no boundary.
• Enclosure: A closed surface encloses a volume of space, whereas an open surface does not enclose any space.
• Use in Gauss's Law: A closed surface is used in **Gauss's law** for calculating electric flux or magnetic flux. An open surface is used in calculating the flux passing through it from a field.

Conclusion:

In summary, an open surface does not enclose any space and has a boundary, while a closed surface completely encloses a space with no boundary. These surfaces are essential in different physical concepts like flux calculations and Gauss's law.

Least Distance from the Central Maximum Where Bright Fringes Coincide

Given:

• Two wavelengths: and
• Distance from slits to screen:
• Separation between slits:

Formula for Bright Fringe Position:

The position of the -th bright fringe for wavelength is given by:

Condition for Coincidence of Bright Fringes:

For the bright fringes due to both wavelengths to coincide, their positions must be the same: Simplifying:

Solution Example:

• Let (for light 1)
• Let (for light 2)
• Let
• Let

Step 1: Calculate the Ratio of and :

Thus, the smallest values of and .

Step 2: Calculate the Coincidence Distance:

The least distance where the fringes coincide is:

Conclusion:

The least distance from the central maximum where the bright fringes due to both wavelengths coincide is **3.6 mm**.