SATHEE: electric-charges-and-fields Question 49

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Question: Q. 8. A point charge $+ Q$ is placed at the centre $O$ of an uncharged hollow spherical conductor of inner radius ’ $a$ ’ and outer radius ’ $b$ ‘. Find the following:

(i) The magnitude and sign of the charge induced on the inner and outer surface of the conducting shell.

(ii) The magnitude of electric field vector at a distance (i) $r = \frac{a}{2}$ , and (ii) $r = 2 b$ , from the centre of the shell.

A&E [CBSE SQP 2017-18]

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Solution:

Ans. (i) As the electrostatic field inside a conductor is zero, using Gauss’s law, charge on the inner surface of the shell $= - Q$

Charge on the outer surface of the shell $= + Q 1 / 2$

(ii) To show using Gauss’s law

Expression for electric field for radius, $r = \frac{a}{2}$ :

$E = \frac{1}{4 π ε_{0}} \cdot \frac{4 Q}{a^{2}}$

Expression for electric field for radius, $r = 2 b$ :

$\begin{matrix} (1) & E = \frac{1}{4 π ε_{0}} \cdot \frac{Q}{b^{2}} \end{matrix}$

[CBSE Marking Scheme, 2017]

Detailed Answer :

(i) Charge placed at the centre of a shell is $+ Q$ . Hence, a charge of magnitude $- Q$ will be induced to the inner surface of the shell. Therefore, total charge on the inner surface of the shell is $- Q$ .

A charge of $+ Q$ is induced on the outer surface of the shell, so a charge of magnitude $Q$ is placed on the outer surface of the shell.

(ii) The electric flux is electric field times the area of spherical surface

$ϕ = E A = 4 E π r^{2} = \frac{Q}{ε_{0}}$

So electric field outside the sphere $(r > R)$ is seen to be identical to that of a point charge $Q$ at center of sphere.

$E = \frac{Q}{4 π ε_{0} r^{2}}$

Magnitude of electric field vector at a distance

$\begin{aligned} r & = \frac{a}{2} E & = \frac{1}{4 π ε_{0}} \cdot \frac{4 Q}{a^{2}} \end{aligned}$

Magnitude of electric field vector at a distance

$\begin{aligned} r = 2 b & E = \frac{1}{4 π ε_{0}} \cdot \frac{4 Q}{4 b^{2}} = \frac{1}{4 π ε_{0}} \cdot \frac{Q}{b^{2}} 1 \end{aligned}$

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विद्युत आवेश और क्षेत्र

Detailed Answer :

विषयसूची

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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