# Solving Gauss's Law for Electric Field at y=0.2m & y=0.6m

• carloqthegrea
In summary, the problem involves finding the net electric field at two points on the y-axis due to two long, uniform lines of charge with given charge per unit length values. This can be solved using Gauss' Law or by considering the contribution of each charge element as a point charge and using integration.
carloqthegrea

## Homework Statement

A very long uniform line of charge has charge per unit length 4.8*10^-6 Coulomb/meter and lies along the x-axis. A second long uniform line of charge has charge per unit length -2.4*106-6 coulomb/meter and is parallel to the x-axis at y=.400 meters. What is the net electric field (magnitude and direction) at the following points on the y-axis: a). y=0.200 metes b). y=0.600 meters?

not sure

## The Attempt at a Solution

Relevant Equation:

$$\vec{E}=K\int_{-\infty}^{\infty}\frac{dq}{r^2}\hat{r}$$

Express dq and r in terms of a single varible and use symmetry.

I still don't know what to do can you give me more details.

Draw a picture of one line charge along the x axis. Choose a differential element of charge, dq, somewhere on the x-axis away from the origin and choose a point on the y axis. Draw a line connecting the point on the y-axis to the charge element thus forming a right triange. This is the distance r. Use the properties of a right triangle to express r in terms of the x coordinate and the y coordinate (the y coordinate will be a constant so call it h for height above the x axis). Also, express dq in terms of x and the charge density. Note, if you choose the charge element on the other side of the y-axis the same distance from this axis the horizontal components of the E fields from the two charges cancel so you only have to determine the vertical component. Use intergration to sum up all the vertical components contributed from all the charge elements. Apply this method to the other line of charge then sum the two resultant E fields at the point h.

chrisk said:
Draw a picture of one line charge along the x axis. Choose a differential element of charge, dq, somewhere on the x-axis away from the origin and choose a point on the y axis. Draw a line connecting the point on the y-axis to the charge element thus forming a right triange. This is the distance r. Use the properties of a right triangle to express r in terms of the x coordinate and the y coordinate (the y coordinate will be a constant so call it h for height above the x axis). Also, express dq in terms of x and the charge density. Note, if you choose the charge element on the other side of the y-axis the same distance from this axis the horizontal components of the E fields from the two charges cancel so you only have to determine the vertical component. Use intergration to sum up all the vertical components contributed from all the charge elements. Apply this method to the other line of charge then sum the two resultant E fields at the point h.

I took "long line of charge" to essentially mean an infinitely long line of charge. In that case, Gauss' Law is much easier than using your method.

@carloqthegrea---Just use a Gaussian cylinder of length 'L' and radius 'r', with the x-axis as its axis of symmetry. What is the charge enclosed by the cylinder? Can you reason that $$\vec{E}$$ is axial and uniform over your Gaussian surface? What does that make the flux through the surface?

The problem states to find the net E field at a point on the y axis. This is the contribution of each charge element considered as a point charge. Guass's Law does not apply to this problem e.g. how would one find the net electric field created by three point charges at a particular point in space?

chrisk said:
The problem states to find the net E field at a point on the y axis. This is the contribution of each charge element considered as a point charge. Guass's Law does not apply to this problem e.g. how would one find the net electric field created by three point charges at a particular point in space?

You don't have three point charges--you have an infinitely long continuous charge distribution. Gauss' Law works just fine and this is a common introductory level application of Gauss' Law. See for example, 'Griffiths Introduction to Electrodynamics' 3rd ed. Prob. 2.13.

It can be done both ways. Sorry, gabbagabbahey. See the below link.

http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/ElectricForce/LineChargeDer.html

Last edited by a moderator:
sweet that was plenty of help, I solved the problem thanks guys.

## 1. How do I solve Gauss's Law for Electric Field at a specific distance from a charge?

To solve Gauss's Law for Electric Field at a specific distance, you will need to use the formula E = k * (q/r^2), where E is the electric field, k is the Coulomb's constant, q is the charge, and r is the distance from the charge. Plug in the values for the charge and distance to calculate the electric field at that point.

## 2. What are the units of the electric field in Gauss's Law?

The units of the electric field in Gauss's Law are Newtons per Coulomb (N/C). This represents the force per unit charge exerted by an electric field.

## 3. How does the electric field change as the distance from the charge increases?

According to Gauss's Law, the electric field decreases as the distance from the charge increases. This is because the electric field is inversely proportional to the square of the distance from the charge.

## 4. Can Gauss's Law be used to calculate electric field at any point in space?

Yes, Gauss's Law can be used to calculate the electric field at any point in space. However, it is important to note that this law only applies to static electric fields and does not take into account changing magnetic fields.

## 5. How do I determine the direction of the electric field using Gauss's Law?

The direction of the electric field can be determined using the principle of superposition. Essentially, you will need to consider the direction of the electric field created by each charge individually and then add them together to determine the overall direction of the electric field at the specific point in space.

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