The document describes the electric field created by a uniformly charged ring at a point P located a distance X from the ring's center. It is shown that the perpendicular components of the electric field created by each segment of the charged ring cancel out, leaving only the parallel component Eχ. Therefore, the total electric field E at P must lie along the X-axis.

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a
Uniform Charged ring has radius length a
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a
Uniform Charged ring has radius length a
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Total charge of ring is Q
Uniform Charged ring has total charge Q

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a
Uniform Charged ring has radius length a
+
Total charge of ring is Q
Uniform Charged ring has total charge Q
P
P is a point lying a distance X from center C
X

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a
Uniform Charged ring has radius length a
+
Total charge of ring is Q
Uniform Charged ring has total charge Q
P
P is a point lying a distance X from center
X
Solution:

6. +
+
+
+
+
+
+
a
Uniform Charged ring has radius length a
+
Total charge of ring is Q
Uniform Charged ring has total charge Q
P
P is a point lying a distance X from center
X
Solution:
Take a small charged segment from
charged ring
dq is charge of small segment
dq

7. +
+
+
+
+
+
+
a
Uniform Charged ring has radius length a
+
Total charge of ring is Q
Uniform Charged ring has total charge Q
P
P is a point lying a distance X from center
X
Solution:
Take a small charged segment from
charged ring
dq is charge of small segment
r is distance between small segment and P
dq
r

8. +
+
+
+
+
+
+
a
Uniform Charged ring has radius length a
+
Total charge of ring is Q
Uniform Charged ring has total charge Q
P
P is a point lying a distance X from center
X
Solution:
Take a small charged segment from
charged ring
dq is charge of small segment
r is distance between small segment and P
r² = a² + X²
dq
r

9. +
+
+
+
+
+
+
a
Uniform Charged ring has radius length a
+
Total charge of ring is Q
Uniform Charged ring has total charge Q
P
P is a point lying a distance X from center
X
Solution:
Take a small charged segment from
charged ring
dq is charge of small segment
r is distance between small segment and P
r² = a² + X²
 dE is electric field at P due to small
segment
dE = k dq /r²
dq
r

10. +
+
+
+
+
+
+
a
Uniform Charged ring has radius length a
+
Total charge of ring is Q
Uniform Charged ring has total charge Q
P
P is a point lying a distance X from center
X
Solution:
Take a small charged segment from
charged ring
dq is charge of small segment
r is distance between small segment and P
r² = a² + X²
 dE is electric field at P due to small
segment
dE = k dq /r²
dq
r
dE

11. +
+
+
+
+
+
+
a
Uniform Charged ring has radius length a
+
Total charge of ring is Q
Uniform Charged ring has total charge Q
P
P is a point lying a distance X from center
X
Solution:
Take a small charged segment from
charged ring
dq is charge of small segment
r is distance between small segment and P
r² = a² + X²
 dE is electric field at P due to small
segment
dE = k dq /r²
dq
r
dE
Ө
Ө X - axis
Y - axis

12. +
+
+
+
+
+
+
a
Uniform Charged ring has radius length a
+
Total charge of ring is Q
Uniform Charged ring has total charge Q
P
P is a point lying a distance X from center
X
Solution:
Take a small charged segment from
charged ring
dq is charge of small segment
r is distance between small segment and P
r² = a² + X²
 dE is electric field at P due to small
segment
dE = k dq /r²
dq
r
dE
Ө dE cos Ө
Ө X - axis
Y - axis

13. +
+
+
+
+
+
+
a
Uniform Charged ring has radius length a
+
Total charge of ring is Q
Uniform Charged ring has total charge Q
P
P is a point lying a distance X from center
X
Solution:
Take a small charged segment from
charged ring
dq is charge of small segment
r is distance between small segment and P
r² = a² + X²
 dE is electric field at P due to small
segment
dE = k dq /r²
dq
r
dE
Ө dE cos Ө = dEᵪ
Ө X - axis
Y - axis

14. +
+
+
+
+
+
+
a
Uniform Charged ring has radius length a
+
Total charge of ring is Q
Uniform Charged ring has total charge Q
P
P is a point lying a distance X from center
X
Solution:
Take a small charged segment from
charged ring
dq is charge of small segment
r is distance between small segment and P
r² = a² + X²
 dE is electric field at P due to small
segment
dE = k dq /r²
dq
r
dE
Ө dE cos Ө = dEᵪ
Ө X - axis
dE
Y - axis

15. +
+
+
+
+
+
+
a
Uniform Charged ring has radius length a
+
Total charge of ring is Q
Uniform Charged ring has total charge Q
P
P is a point lying a distance X from center
X
Solution:
Take a small charged segment from
charged ring
dq is charge of small segment
r is distance between small segment and P
r² = a² + X²
 dE is electric field at P due to small
segment
dE = k dq /r²
dq
r
dE
Ө dE cos Ө = dEᵪ
Ө X - axis
dE = Zero
Y - axis

16. +
+
+
+
+
+
+
a
Uniform Charged ring has radius length a
+
Total charge of ring is Q
Uniform Charged ring has total charge Q
P
P is a point lying a distance X from center
X
Solution:
Take a small charged segment from
charged ring
dq is charge of small segment
r is distance between small segment and P
r² = a² + X²
 dE is electric field at P due to small
segment
dE = k dq /r²
 dEᵪ = dE cos Ө = (k dq /r²) . X/r
dq
r
dE
Ө dE cos Ө = dEᵪ
Ө X - axis
dE = Zero
Y - axis

17. +
+
+
+
+
+
+
a
Uniform Charged ring has radius length a
+
Total charge of ring is Q
Uniform Charged ring has total charge Q
P
P is a point lying a distance X from center
X
Solution:
Take a small charged segment from
charged ring
dq is charge of small segment
r is distance between small segment and P
r² = a² + X²
 dE is electric field at P due to small
segment
dE = k dq /r²
 dEᵪ = dE cos Ө = (k dq /r²) . X/r
dq
r
dE
Ө dE cos Ө = dEᵪ
Ө X - axis
dE = Zero
Y - axis
NB: r² = a² + X²

18. +
+
+
+
+
+
+
a
Uniform Charged ring has radius length a
+
Total charge of ring is Q
Uniform Charged ring has total charge Q
P
P is a point lying a distance X from center
X
Solution:
Take a small charged segment from
charged ring
dq is charge of small segment
r is distance between small segment and P
r² = a² + X²
 dE is electric field at P due to small
segment
dE = k dq /r²
 dEᵪ = dE cos Ө = (k dq /r²) . X/r
dq
r
dE
Ө dE cos Ө = dEᵪ
Ө X - axis
dE = Zero
Y - axis
NB: r² = a² + X²
and r = (a² + X²)²

20. Explain:
The total electric field “E” of uniform charged ring of radius “a” at point “P” is electric field on
the X-axis “Eᵪ”
Or
In case of uniform charged ring The Resultant of electric field at point “P” must lie along the
X-axis (i.e. Eᵪ is the resultant field)
Answer:
The perpendicular component of the electric field created by any charge element is canceled
by the perpendicular component of the electric field created by an element on the opposite
side of charged ring.
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a
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P
dq₁
dE₁
X - axis
Y - axis
dq₂
dE₂
Ө
dE₁ sin Ө
dE₂ sin Ө
Ө
Ө
Ө

21. Suppose a negative charge is placed at the center of the ring and
displaced slightly by a distance x << a along X-axis. What type of
motion dose it exhibit?

22. Electric Field Lines: