Finding the Electric Force on a Rod by Linear Charge

[thiefjack]

Homework Statement

An infinitely long, uniformly charged straight line has linear charge density λ1 coul/m. A straight rod of length 'b' lies in the plane of the straight line and perpendicular to it, with its enared end at distance 'a' from the line. The charge density on the rod varies with distance 'y', measured from the lower end, according to λ(on rod) = (λ2*b)/(y+a), where λ2 is a constant. Find the electrical force exerted on the rod by the charge on the infinite straight line, in the λ1, λ2, a, and b, and constants like ε0.

See attachment.

Homework Equations

F = [1/(4πε0)] * [(q1 * q2)/(r^2)]

The Attempt at a Solution

My idea is to first find the the integral of all the forces on an arbitrary charge on the rod. Then integrate the sum of that force as you go up along the rod with the different charge density.

I'm just a bit confused as to how I can utilize λ2.

If we take the line charge as the x-axis and the rod as the y axis, then the distance between two charges is r = sq[x^2 + (a+y)^2]

 

[SammyS]

Homework Statement

An infinitely long, uniformly charged straight line has linear charge density λ₁ coul/m. A straight rod of length 'b' lies in the plane of the straight line and perpendicular to it, with its nearer end at distance 'a' from the line. The charge density on the rod varies with distance 'y', measured from the lower end, according to λ(on rod) = (λ₂*b)/(y+a), where λ₂ is a constant. Find the electrical force exerted on the rod by the charge on the infinite straight line, in the λ₁, λ₂, a, and b, and constants like ε₀.

See attachment.

Homework Equations

F = [1/(4πε0)] * [(q₁ * q₂)/(r²)]

The Attempt at a Solution

My idea is to first find the the integral of all the forces on an arbitrary charge on the rod. Then integrate the sum of that force as you go up along the rod with the different charge density.

I'm just a bit confused as to how I can utilize λ₂.

If we take the line charge as the x-axis and the rod as the y axis, then the distance between two charges is r = sqrt[x² + (a+y)²]

Hello thiefjack. Welcome to PF !

Do you know the expression for the Electric field due to an infinitely long line of charge?

 

[thiefjack]

Hi! Now that you remind me, I do remember deriving it awhile ago. Completely forgot about it.

E = (2kλ/y)j

Thanks for the reminder.

 

[SammyS]

So, for any small length, dy, of the rod, with a charge of dq, the force is (E)(dq).