Findng electric force for line of charge

[lonewolf219]

Homework Statement

Find the force acting on an electron located "d" distance from the midpoint of a line of charge, length "L", located on the x axis. The line of charge is positive.

Homework Equations

F(e)=kq1q2/r^2

The Attempt at a Solution

λ=linear density, here the charge is uniform.
So, dq=λdx I think?
The distance is the variable that is changing, so we should integrate the dx and the limits should be over the length of the line?

k, e(electron) and λ are constant and can be brought out of integral... if the origin is the midpoint, then the limits of integration are -L/2 to L/2? The solution the book gives is the following:

4keλL/(4d^2-L^2)

I don't know where I'm going wrong with this equation, maybe I have the wrong r value?
Is r=(d-x)^2 when dx is along the positive x axis?
Any tips or perspectives would be great!

 

[tiny-tim]

hi lonewolf219!

So, dq=λdx I think?

yes

Is r=(d-x)^2 when dx is along the positive x axis?

how can a distance equal a distance squared?

(and don't forget that force is a vector, so you'll need to integrate the component )

 

[lonewolf219]

Ah, yes, that's why I prefer energy! The electron is also along the x axis, so the angle between the line and the electron is 0, so cos(0)=1?

Using Coulomb's formula, where denominator "r" is squared. The distance between the electron and a dx element of the line (?) can be represented by what? Is it d-x, where we don't know the value of x? Or d+x?

 

[tiny-tim]

Using Coulomb's formula, where denominator "r" is squared. The distance between the electron and a dx element of the line (?) can be represented by what? Is it d-x, where we don't know the value of x? Or d+x?

ah, i think the question means that the electron is on the y axis

The electron is also along the x axis, so the angle between the line and the electron is 0, so cos(0)=1?

see above

 

[asteriatic]

I would approach this problem by first understanding the physical concept behind the calculation. The electric force acting on an electron is given by Coulomb's Law, which states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. In this case, we have a positive line of charge and an electron located at a distance "d" from the midpoint.

To find the electric force, we can use the equation F(e) = kq1q2/r^2, where k is the Coulomb's constant, q1 and q2 are the charges, and r is the distance between them. In our case, q1 is the charge of the electron, which is a fundamental constant, and q2 is the charge of the line of charge, which is given by the linear density λ and the length of the line L.

Next, we need to find the distance r between the electron and the line of charge. Since the line of charge is located on the x axis, we can use the Pythagorean theorem to calculate the distance. The distance r is given by r = √(d^2 + x^2), where x is the distance from the midpoint of the line of charge.

Now, we can set up an integral to take into account the changing distance x along the line of charge. The limits of integration will be from -L/2 to L/2, as the line of charge extends from -L/2 to L/2. The integrand will then be the force equation F(e) multiplied by the linear density λ and the infinitesimal distance dx.

After integrating, we will get the following equation:

F = ∫ k(e)(λdx)/√(d^2 + x^2)^2

Simplifying this equation will give us the solution provided by the book: 4k(e)λL/(4d^2 - L^2).

If you are still having trouble understanding the solution, I would suggest breaking down the problem into smaller steps and understanding each concept before moving on to the next. It may also be helpful to consult with your professor or a tutor for further clarification.