Finding an expression for Electric Potential?

In summary, the electric potential at the center of a semicircular rod with charge Q and radius R can be calculated using the formula V = kQ/R, where k is a constant and R is the radius. This can be derived by integrating over the angle θ using polar coordinates, with limits of integration from 0 to π. The extra R in the denominator is cancelled out by using rdθ instead of dx in the integrand.
  • #1
PolarBee
14
1

Homework Statement


There is a thin rod with charge Q that has been bent into a semicircle with radius R. Find an expression for the electric potential at the center.

Radius = R
Charge = Q

Homework Equations


V = ∫(k * dq)/r

The Attempt at a Solution


dq = λdx
λ = Q/L
L = pi * r

V = kQ / LR ∫ dx = kQ / pi * r^2 ∫ [3pi/2 - pi/2]
V = kQ/R^2

But the answer should be kQ/R. How did the other R get cancelled?
 
Physics news on Phys.org
  • #2
What are your limits of integration? You should use polar coordinates and integrate over an angle.
 
  • #3
I integrated from pi/2 to 3pi/2. Which came out to be just pi and that canceled out the pi from L = pi * r.
 
  • #4
PolarBee said:
I integrated from pi/2 to 3pi/2. Which came out to be just pi and that canceled out the pi from L = pi * r.
Your integrand has a dx in it. What did you do with it? To integrate over an angle you need a dθ in the integrand.
 
  • #5
Oh I see. So dx should be rdθ instead for polar coordinates right, without it I can't integrate from pi/2 to 3pi/2?
 
  • Like
Likes MUUKGIZ
  • #6
Correct. That cancels the extra R in the denominator. Also, note that the beginning angle is arbitrary. A semicircle is a semicircle. You might as well integrate from 0 to π.
 

Related to Finding an expression for Electric Potential?

1. What is Electric Potential?

Electric potential is a measure of the potential energy of a charged particle in an electric field. It is also known as voltage and is measured in volts (V).

2. How is Electric Potential different from Electric Field?

Electric potential is a scalar quantity, while electric field is a vector quantity. This means that electric potential only has magnitude, while electric field has both magnitude and direction.

3. How do you calculate Electric Potential?

The formula for electric potential is V = kQ/r, where V is the electric potential, k is a constant, Q is the charge of the particle, and r is the distance between the particle and the point where the electric potential is being measured.

4. What is the relationship between Electric Potential and Electric Field?

Electric potential is related to electric field by the formula E = -∇V, where E is the electric field and ∇V is the gradient of the electric potential. This means that electric field points in the direction of decreasing electric potential.

5. How is the concept of Electric Potential used in real-life applications?

Electric potential is used in various applications, such as in electronic circuits, power distribution systems, and electrostatic precipitators. It is also an important concept in understanding the behavior of charged particles in fields, such as in particle accelerators and mass spectrometers.

Similar threads

  • Introductory Physics Homework Help
Replies
2
Views
548
Replies
22
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
995
  • Introductory Physics Homework Help
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
23
Views
519
  • Introductory Physics Homework Help
Replies
6
Views
274
  • Introductory Physics Homework Help
Replies
11
Views
2K
  • Introductory Physics Homework Help
Replies
4
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
1K
Back
Top