Electric field of wire and cylinder at one point in space

[uni98]

Homework Statement

Given a cylinder of radius R with volumetric charge density, whose axis coincides with the z axis and a uniformly charged infinite wire that coincides with the y axis. Calculate the force felt by a point charge q placed at point A (R / 3, R / 3, R).

Relevant Equations

Cylinder field inside: ρR^2/2rε

Cylinder field outside: ρr/2ε

Field generated by the wire: λ/2πr

I can calculate the fields generated by the cylinder and the wire but I don't know how to calculate their vector sum to evaluate it at point A.
Cylinder field inside: ρR^2/2rε
Cylinder field outside: ρr/2ε
Field generated by the wire: λ/2πr
I should break the fields into components but I don't know how to proceed. Even if you know some sites on which I can find information about it, that's fine. I hope for your help, I really need it. Thank you in advance.

 

[kuruman]

I should break the fields into components but I don't know how to proceed.

Make a drawing with x, y and z axes. Then draw point A. You know the magnitude of the field at A due to the wire alone. Draw an arrow representing that field. In what direction is that arrow?

Once you have that, draw a second arrow at A due to the cylinder alone. In what direction is that? Add the arrows as vectors.

If you get stuck, be sure to post what you have done.

 

[uni98]

Make a drawing with x, y and z axes. Then draw point A. You know the magnitude of the field at A due to the wire alone. Draw an arrow representing that field. In what direction is that arrow?

In the meantime, excuse me if the drawing is not very clear. I represented the thread with the purple color and point A is green. I have represented the vector of the electric field generated by the wire towards point A and its components which are along the z and x axis.

Once you have that, draw a second arrow at A due to the cylinder alone. In what direction is that? Add the arrows as vectors.

I proceeded in the same way with the cylinder and I represented the components of the electric field towards A, the components are along the x and y axis.

At this point, I'm probably missing some basic trigonometry concepts that I should review. I should break down each of the electric field vectors into their x, y, z components. If you could tell me how you would break them down, I can try to guess the logic.

 

[Delta2]

You might have to see this
https://en.wikipedia.org/wiki/Vecto...cal_coordinates#Cylindrical_coordinate_system

in order to check how to convert between the components of cylindrical coordinates $E_r,E_{\phi},E_z$ and the components of cartesian coordinates $E_x,E_y,E_z$ (ok it is obvious that the $E_z$ components are the same).

Extra caution is needed when you will convert the electric field from the infinite wire because there the y-axis is essentially the z-axis of its cylindrical coordinate system so the roles of y and z get swapped.

Once you have the fields in the cartesian components of the common cartesian coordinate system, then it is a straightforward task to add them, you just add them component wise,i.e. $E_x+E'_x$ for the x-component of the total E-field, (where $E_x$ the x-component of the field from the cylinder and $E'_x$ the x-component of the field from the wire).

 

[Delta2]

In order to help a bit more the link I gave you is about how to convert between the coordinates , and how to convert between the unit vectors. If you understand the conversion of the unit vectors then it is a straightforward task to convert between the components.
For example if
$$\hat r=\cos\phi\hat x+\sin\phi\hat y$$ then the it would be $$E_r\hat r=E_r\cos\phi\hat x+E_r\sin\phi\hat y$$ so it would be $$E_x=E_r\cos\phi+...$$ and $$E_y=E_r\sin\phi+...$$ where i put +... because there is additional contribution from the $E_{\phi}$ component but ok this component is actually zero in this problem

 

[kuruman]

I proceeded in the same way with the cylinder and I represented the components of the electric field towards A, the components are along the x and y axis.

You will not get very far unless your drawing is correct and you have a correct conceptual understanding of what's going on.

If you want to find the superposition of two fields, one from the wire and one from the cylinder, at point A, you need to represent them as two arrows both of which have their tails at point A. That's what you need to add.

First make a drawing in the xz plane. Draw a dot for the wire coming out of the screen. Add point A. Draw the single arrow for the field due to the wire. Define some angle relative to the z-axis and find the components. Remember SOH-CAH-TOA. The electric field is the hypotenuse. Do that first and then we'll worry about the cylinder.

On edit: You need to swap the labels for what's inside and what's outside for the fields due to the cylinder. Also, your expression for the field due to the wire needs a factor of ε₀ in the denominator.