The linear charge density on a dielectric ring of radius R is varying with θ as λ$$=$$λ$$0cos$$⁡θ$$2$$, where λ$$0$$ is constant. The potential at the centre O of ring [in volt] $$is_____________$$

Question

The linear charge density on a dielectric ring of radius R is varying with θ as λ$$=$$λ$$0cos$$⁡θ$$2$$, where λ$$0$$ is constant. The potential at the centre O of ring [in volt] $$is_____________$$

Infinity Learn · Accepted Answer

The charge on the infinitesimal elements of arc which subtend an angle d $θ$ at the centre of the ring

$d Q = λ R d θ = λ_{0} \cos \frac{θ}{2} R d θ$

Potential at the centre of ring due to charge dQ

$\begin{array}{l} d V = \frac{1}{4 π ϵ_{0}} \frac{d Q}{R} = \frac{λ_{0} \cos \frac{θ}{2} \cdot R d θ}{4 π \in_{0} R} \\ V = \int d V \Rightarrow V = \frac{λ_{0}}{4 π ϵ_{0}} \int_{0}^{2 π} \cos \frac{θ}{2} d θ \\ = \frac{λ_{0}}{4 π ϵ_{0}} {[\frac{\sin \frac{θ}{2}}{\frac{1}{2}}]}_{0}^{2 π} = 0 V \end{array}$

Electrostatic potential

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The linear charge density on a dielectric ring of radius R is varying with $θ$ as $λ = λ_{0} \cos \frac{θ}{2}$ , where $λ_{0}$ is constant. The potential at the centre O of ring [in volt] is_____________

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Electrostatic potential

Remember concepts with our Masterclasses.

The linear charge density on a dielectric ring of radius R is varying with θ as λ=λ0cos⁡θ2, where λ0 is constant. The potential at the centre O of ring [in volt] is_____________

The charge on the infinitesimal elements of arc which subtend an angle dθ at the centre of the ringdQ=λRdθ=λ0cos⁡θ2RdθPotential at the centre of ring due to charge dQdV=14πϵ0dQR=λ0cos⁡θ2⋅Rdθ4π∈0RV=∫dV⇒V=λ04πϵ0∫02π cos⁡θ2dθ=λ04πϵ0sin⁡θ21202π=0V

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The linear charge density on a dielectric ring of radius R is varying with $θ$ as $λ = λ_{0} \cos \frac{θ}{2}$ , where $λ_{0}$ is constant. The potential at the centre O of ring [in volt] is_____________