Gauss's Law and Long wire.

[K3nt70]

Homework Statement

Calculate the electric field at a point 2.00 cm perpendicular to the midpoint of a 1.94 m long thin wire carrying a total charge of 4.78 uC.

Hint: You could integrate BUT if the wire is very long compared to the distance from the wire to where you are calculating the electric field, then the electric field will be radial and Gauss's law might be easier.

Homework Equations

E = $$\frac{kq}{r^2}$$

where k = 9E9
q = 4.78E-6
r = 0.02 m

E = $$\frac{kQx}{(x^2 + a^2)^(3/2)}$$

The Attempt at a Solution

First i tried to use the second formula where i made the wire into a circle and used 0.02m as the radius. That didnt work, so i used gauss's formula as the hint suggests, which also didnt work. Suggestions?

 

[K3nt70]

Nevermind, i solved it.

 

[Moetasim]

I would suggest approaching this problem by first understanding the concept of Gauss's Law. Gauss's Law states that the electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of free space. In simpler terms, it means that the electric field at a point is directly proportional to the amount of charge enclosed by a surface and inversely proportional to the distance from the point to the charge.

In this problem, the wire is very long compared to the distance from the wire to the point where we are calculating the electric field. Therefore, we can consider the wire as a long, straight line and use Gauss's Law to simplify the calculation.

To do so, we can use a Gaussian surface in the shape of a cylinder with the wire passing through its center. The electric field will be constant along the curved surface of the cylinder and perpendicular to it. The only contribution to the electric flux will come from the ends of the cylinder, where the electric field lines are parallel to the surface. This means that the electric flux through the curved surface will be zero, and we only need to consider the flux through the ends.

Since the charge is distributed along the wire, we can use the linear charge density, λ, which is defined as the charge per unit length of the wire. In this case, we can calculate the linear charge density by dividing the total charge by the length of the wire.

λ = Q/L = 4.78E-6 C / 1.94 m = 2.465E-6 C/m

Now, using Gauss's Law, we can calculate the electric field at a point 2.00 cm perpendicular to the midpoint of the wire.

E = \frac{λ}{ε_0} = \frac{2.465E-6 C/m}{8.85E-12 C^2/Nm^2} = 2.78E6 N/C

Therefore, the electric field at the given point is 2.78E6 N/C, directed radially away from the wire.

In conclusion, by using Gauss's Law and understanding the concept of electric flux, we can simplify the calculation and accurately determine the electric field at a point near a long wire carrying a total charge.