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PhysicsElectrostaticsRelation between field & potentialMedium2 minQB

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A non conducting sphere having relative permittivity unity $(ε_{r} = 1)$ has radius R and have a non uniform charge distribution. The volume charge density $ρ (r)$ varies with radial distance r as per the following graph. Here $ρ_{0}$ is a constant.\\

Consider an isolated sphere (as explained in paragraph). If $E$ is electric field at any point because of sphere and if C is centre of sphere and S is surface of sphere find the value of $\int_{S}^{C} E \cdot d r$ .

Options:

Answer:

Solution:

$Q_{in} = \int_{0}^{r} \frac{ρ _{0} r}{R} 4 π r^{2} d r = \frac{ρ _{0} π r ^{4}}{R}$ , so $E_{in} = \frac{k Q _{in}}{r ^{2}} = \frac{ρ _{0} r ^{2}}{4 ε _{0} R}$ .

$\int_{S}^{C} E \cdot d r = \int_{R}^{0} \frac{ρ _{0} r ^{2}}{4 ε _{0} R} d r = - \frac{ρ _{0} R ^{2}}{12 ε _{0}}$ - (A)

Stream:JEESubject:PhysicsTopic:ElectrostaticsSubtopic:Relation between field & potential

⏱ 2mℹ️ Source: QB

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