# Electric field of a line of charge and direction

• Ahmed A
In summary, the problem involves a nonconducting plastic wire with a charge density of +175 nC/m distributed uniformly along its length. The wire is lying horizontally on a table top and the task is to find the magnitude and direction of the electric field it produces at a point 6.00 cm directly above its midpoint. Using the given equations and attempting a solution, a value of 3.03 * 10^4 N/C was obtained, which differs from other sources claiming the correct answer to be 4.28 * 10^4 N/C. The solution shown does not seem to have any errors.
Ahmed A

## Homework Statement

A straight nonconducting plastic wire 8.50 cm long carries a charge density of +175 nC/m distributed uniformly along its length. It is lying on a horizontal table top. Find the magnitude and direction of the electric field this wire produces at a point 6.00 cm directly above its midpoint.

## Homework Equations

E = $\frac{1}{4πε_0}$ * $\frac{Q}{r^2}$
λ = 175 nC/m
a = 4.25 cm
y = 6.00 cm

## The Attempt at a Solution

I made a diagram with the rod along the x-axis, with the y-axis intersecting it at the center. The endpoints of the rod are ±a where a=4.25cm.

dE = $\frac{1}{4πε_0}$ * $\frac{λdx}{x^2+y^2}$
dEy = dE sin(θ)
sin(θ) = $\frac{y}{\sqrt{x^2+y^2}}$

dEy = $\frac{λy}{4πε_0}$ * $\frac{dx}{(x^2+y^2)^\frac{3}{2}}$
Ey = $\frac{λy}{4πε_0}$ * $\int^{a}_{-a} \frac{dx}{(x^2+y^2)^\frac{3}{2}}$
Ey = $\frac{λa}{2πε_{0}y\sqrt{x^2+y^2}}$

The result I got after plugging all the numbers in is E = 3.03 * 104 N/C but other sources online say the correct answer is 4.28 * 104 N/C. What did I do wrong?

Last edited:
Ahmed Elwakeel
I don't see a problem with what you've done. (Although you did forget to replace the 'x2' within the radical of the final line with 'a2' -- no doubt a minor oversight).

## 1. What is the formula for calculating the electric field of a line of charge?

The electric field of a line of charge can be calculated using the formula E = k * λ / r, where k is the Coulomb's constant, λ is the charge density of the line, and r is the distance from the line of charge.

## 2. How does the direction of the electric field change as you move away from the line of charge?

The electric field of a line of charge is always directed radially outward or inward from the line. As you move away from the line of charge, the direction of the electric field becomes more perpendicular to the line.

## 3. Can the electric field of a line of charge be negative?

Yes, the electric field of a line of charge can be negative. This occurs when the line of charge has a negative charge density, resulting in an electric field that is directed inward towards the line.

## 4. How does the electric field of a line of charge compare to the electric field of a point charge?

The electric field of a line of charge is similar to the electric field of a point charge in that it follows an inverse square law, but the direction of the electric field is different. The electric field of a point charge is radial, while the electric field of a line of charge is perpendicular to the line.

## 5. What is the significance of the direction of the electric field of a line of charge?

The direction of the electric field of a line of charge is important because it determines the direction in which a positive test charge would travel if placed in the electric field. It also helps in understanding the behavior of charged particles in the presence of a line of charge.

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