Stuck again - electric field and circular arc

[PinkFlamingo]

Hi there,

I was hoping that somebody could help me. I'm stuck again!

a rod with (lambda) coulombs of charge per meter of its length has the shape of a circular arc of radius R. The rod subtends an angle (theta). Shpw that the magnitude of the electric field at the centre of the circular arc is:

E=[1/(4*pi*E0)] [(2*lambda)/R] sin (theta/2)

Where R is the radius from the rod to the centre of the electic field

OK, I know that I somehow have to change the formula for the electric field of a rod to get that. The formula for the electric field of a rod is:

E==[1/(4*pi*E0)] [(2*lambda)/r]

where r is the vertical distance from the rod

 

[turin]

Do you know how to integrate?

The electric field for the rod that you gave is for an infinitely long rod. You should first consider whince the formula comes. It is based on Coulomb's Law, superposition, and the definition of the electric field.

To adjust Coulomb's Law, replace the source charge q with dq. This indicates that the charge in Coulomb's Law is just a little piece of some object.

The superposition is the (relatively) complicated issue; it involves integration as well as a cleaver substitution for dq. In general, for a linear charge distribution (a string of charge), you want something of the form: dq = λ(s)ds (where s is the arclength position along the string). I will give you the example for a line of charge along the x-direction: dq = λ(x)dx. Try to figure what it would be for a circular arc. Then, you stick Coulomb's Law with this modification into an integral (over x, for instance). Be careful: the ds may not be the only thing that integrates. You can pull some factors out of the integral, but some must remain as part of the integrand because they depend on the variable of integration.

Then, just divide by the test charge to get the electric field. This can be done in the beginning.

 

[M98Ranger]

to the point where the electric field is measured. But I'm not sure how to incorporate the circular arc shape and the angle (theta) in this formula. Any help would be greatly appreciated!

Hi there,

I can definitely help you with this problem. Let's start by visualizing the circular arc and its relationship to the electric field at the center. The first thing to note is that the electric field is a vector quantity, meaning it has both magnitude and direction. In this case, we are only interested in the magnitude of the electric field, so we can ignore the direction for now.

Next, we need to consider the geometry of the situation. Since the rod is in the shape of a circular arc, we can imagine it as a segment of a larger circle with radius R. The angle (theta) is the central angle of this segment, which is also the angle that the electric field makes with the rod at the center.

Now, let's think about the formula for the electric field of a rod. As you correctly stated, it is E=[1/(4*pi*E0)] [(2*lambda)/r]. This formula tells us that the electric field is directly proportional to the charge per unit length (lambda) and inversely proportional to the distance from the rod (r).

To incorporate the circular arc shape and the angle (theta), we need to modify this formula. Since the electric field is still proportional to the charge per unit length, we can keep that term the same. However, instead of using the distance from the rod (r), we need to use the radius of the circle (R) since that is the distance from the center of the arc to the center of the electric field.

Now we need to consider the angle (theta). We know that the electric field is strongest when it is perpendicular to the rod. In this case, the electric field is perpendicular when the angle (theta) is 90 degrees. As the angle decreases, the electric field becomes weaker. This relationship can be described by the sine function, where the magnitude of the electric field is equal to the maximum value (2*lambda)/R multiplied by the sine of the angle (theta/2). Putting all of this together, we get the final formula:

E=[1/(4*pi*E0)] [(2*lambda)/R] sin (theta/2)

I hope this helps you understand how to approach this problem. Good luck!

 

[M98Ranger]

to the point where the electric field is being measured.

First of all, don't worry about feeling stuck - that's a normal part of the learning process! Let's break down the problem and see if we can figure it out together.

So, we have a circular arc shaped rod with a charge density of lambda coulombs per meter. The first thing we can do is try to visualize the situation. Can you draw a diagram of the circular arc and label the relevant variables? This might help us to better understand the problem.

Next, let's think about what the formula for the electric field of a rod represents. It tells us the electric field at a point r away from the rod, in terms of the charge density lambda and the distance r. In the case of a circular arc, we are interested in the electric field at the centre of the arc, which is a distance R away from the rod. So, we can substitute R for r in the formula and see what happens.

E=[1/(4*pi*E0)] [(2*lambda)/R]

This gives us the electric field at the centre of the arc. But, we also need to take into account the angle that the arc subtends, as mentioned in the problem. Can you think of a way to incorporate this angle into the formula?

Hint: Think about the geometry of a circular arc and how it relates to the angle theta.

Once you have an idea, try substituting it into the formula and see if it works. Remember to include the sin function in your final formula, as it is related to the angle theta.

I hope this helps and good luck with your problem! Don't hesitate to ask for further clarification if needed.