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A circle of radius a has charge density given by λ = λ0 cos²θ on its circumference. What will be the total charge on the circle? ​

- Se necesita una explicación detallada
- gracias de antemano
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Answer 1

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Tardigrade

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A circle of radius a has charge density given by λ = λ0 cos2 θ on its circumference. What will be the total charge on the circle?

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Q. A circle of radius a has charge density given by λ=λ

0

cos

2

θ on its circumference. What will be the total charge on the circle?

5197 22 Gujarat CETGujarat CET 2008Electric Charges and Fields Report Error

A2πa

11%

BZero

27%

Cπaλ

0

55%

DNone of these

7%

Solution:

Charge density Let PQ be an element of length dl on the circumference of the circle which makes an angle dθ at the centre of the circle.

Then the charge on the element is

dq=λ×dl

⇒dq=λ

0

cos

2

θdl ..(i)

Now, dθ=

a

dl

⇒dl=adθ

Then from Eq. (i)

dq=λ

0

cos

2

θ.adθ

⇒dq=λ

0

acos

2

θdθ ...(ii)

The total charge on the circle [ on integrating Eq. (ii) with limit 0 to 2 π]

q=

0

∫

2π

λ

0

acos

2

θdθ

or q=λ

0

a

0

∫

2π

2

cos2θ+1

dθ

[∵cos

2

θ=

2

cos2θ+1

]

⇒q=

2

λ

0

a

[

2

sin2θ

+θ]

0

2π

⇒q=

2

λ

0

a

[

2

sin4π

+2π−

2

sin0

−0]

⇒q=

2

λ

0

a

[0+2π−0]

⇒q=πaλ

0

Explicación:

Espero que te sirva, saludos