
State Gauss law in electrostatics. Derive an of electric field expression due an infinitely long straight uniformly charged wire. Draw necessary diagrams.
Answer
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Hint: In electrostatics, the Gauss’s law states that the total electric flux over any closed surface is times the total charge that is enclosed by the surface. To find the electric field due to an infinitely long current carrying wire, we need to consider a cylindrical Gaussian surface around it.
Complete Step by Step Solution: The Gauss’s law in electrostatics shows us that there is an important relationship that exists between the total electric flux over any closed surface and the total charge that is enclosed by the surface. It states that the total electric flux over any closed surface is times the total charge that is enclosed by the surface where is the permittivity in free space.
The mathematical form of the Gauss’s law is,
To derive the expression of the electric field due to an infinitely long charged wire, we consider the following diagram,

Here we have considered the Gaussian surface as a cylinder that is surrounding the wire which has a surface charge density . This cylinder has a radius and a length . Let us consider a point P lying on the curved surface of the cylinder. So we need to find the electric field due to the wire at the P.
So, from the Gauss’s law we can write,
In the cylinder there are 3 surfaces present as shown in the figure. So we need to find Gauss's law for all the three surfaces , and .
So we can break the L.H.S. of the integration into three parts as,
Now the direction of the electric field from the wire is radially outwards. So the electric field is perpendicular on the surface and is parallel to the surfaces and . So the angle between the electric field and the surface vectors are for and , and for as the surface vectors are perpendicular to the surfaces. Therefore, we can write, the dot products in form of the magnitudes as,
The value of and
So the first and the third term of the equation vanish.
Therefore, we get
Now since the electric field is constant we can take it out of the integration. So,
The surface integration is over the curved surface of the cylinder. The surface area of the curved surface is
And as the surface charge density of the wire is , so the total charge of the wire for length is,
So substituting the values,
The gets cancelled from both the sides and then taking all the terms except from the LHS to the RHS we get
This is the magnitude of the electric field. The direction of the electric field is radially outwards, so we can write it as
Note: From the formula for the electric field, we can see that it is independent of the length of the wire but depends on the charge density. In case of 3D we need to replace the linear charge density of the wire with its volume charge density.
Complete Step by Step Solution: The Gauss’s law in electrostatics shows us that there is an important relationship that exists between the total electric flux over any closed surface and the total charge that is enclosed by the surface. It states that the total electric flux over any closed surface is
The mathematical form of the Gauss’s law is,
To derive the expression of the electric field due to an infinitely long charged wire, we consider the following diagram,

Here we have considered the Gaussian surface as a cylinder that is surrounding the wire which has a surface charge density
So, from the Gauss’s law we can write,
In the cylinder there are 3 surfaces present as shown in the figure. So we need to find Gauss's law for all the three surfaces
So we can break the L.H.S. of the integration into three parts as,
Now the direction of the electric field from the wire is radially outwards. So the electric field is perpendicular on the surface
The value of
So the first and the third term of the equation vanish.
Therefore, we get
Now since the electric field is constant we can take it out of the integration. So,
The surface integration is over the curved surface of the cylinder. The surface area of the curved surface
And as the surface charge density of the wire is
So substituting the values,
The
This is the magnitude of the electric field. The direction of the electric field is radially outwards, so we can write it as
Note: From the formula for the electric field, we can see that it is independent of the length of the wire but depends on the charge density. In case of 3D we need to replace the linear charge density of the wire with its volume charge density.
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