How to Calculate Total Force on a Charge at the Center of Curvature?

In summary, the conversation is about a homework problem involving a semicircle of positive charge and calculating the total force on a charge at the center of curvature. The charge per unit length is described by a cosine function and the total charge is given. The conversation also includes a formula for electric field and the force on the charge. The final answer given by the book is compared to the one calculated by Mateoguapo, and he is asked to show his work first in the future.
  • #1
mateoguapo327
help! I need help with this homework problem.

--A line of positive charge is formed into a semicircle of radius R=60.0 cm. The charge per unit length along the semicircle is described by the expression [lamb]= [lamb]naught cos [the] . The total charge on the semicircle is 12.0 microcoulombs. Calculate the total force on a charge 3.00 microcoulombs at the center of curvature.--

The figure shows the semicircle with is center at the origin going from 0 to pi. [the] is the angle formed by dragging R clockwise from the positive y axis.

the answer in the book is -0.707Nj and I got -0.526N. Can someone please help me figure out how to properly go about solving this problem?
 
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  • #2


consider the electric field at the center of the semicircle due to a little piece of the semicircle at angle θ. by symmetry, only the vertical component of this field contributes:

dEy = - 1/4πε0 cos θ dq/r2 = - 1/4πε0 cos θ λrdθ/r2 = - 1/4πε0 cos2 θλ0dθ/r

then integrate from π/2 to -&pi/2:

Ey = -λ0/4πε0r∫cos2 θ dθ = -λ0/8ε0r


the force on the charge is the Fy = qEy


we also need to solve for λ0.

Q = ∫dq = ∫λ0rcos θ dθ = 2rλ0

λ0 = Q/2r

finally
Fy = -qQ/16ε0r2


plug in the numbers and you get -0.70588 N.
 
  • #3
Ahem!

Mateoguapo:

https://www.physicsforums.com/showthread.php?s=&threadid=28

In the future, please show your work first.

Thanks,
 
  • #4
oops. my bad.
 

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Electromagnetism is a branch of physics that deals with the study of electric and magnetic fields and their interactions with each other. It is a fundamental force of nature and plays a crucial role in the behavior of matter and energy at both the atomic and macroscopic levels.

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