Derive an expression for the electric potential due to a point charge. A positive charge of magnitude 6 μC is placed in a dielectric medium of dielectric constant 2 . What is the electric potential at a point in the dielectric medium at a distance of 10 cm from the charge ?

Let + q be an isolated point charge situated in air at O . P is a point at a distance r from + q . Consider two points A and B at distances x and x + d x from the point O . The potential difference between A and B is, d V = − E d x .................... 1 The electric field strength at A due to the point charge is, E = 1 4 π ε ∘ q x 2 .................... 2 By substituting equation 2 in equation 1 we get: d V = - 1 4 π ε ∘ q x 2 d x .................... 3 Here, the negative sign indicates that the work is done against the electric force. The electric potential at the point P due to the charge + q is the total work done in moving a unit positive charge from infinity to that point. V = ∫ ∞ r - 1 4 π ε ∘ q x 2 d x V = q 4 π ε ∘ 1 r - 1 ∞ V = 1 4 π ε ∘ q r Hence, the electric potential due to a point charge is 1 4 π ε ∘ q r . Now a point charge of q = 6 μC is placed in a dielectric medium with dielectric constant 2 . Then the electric potential at the given point due to this point charge: V ' = 1 4 π ε ∘ ε r q r V ' = 9 × 10 9 N m 2 C - 2 2 6 × 10 - 6 C 10 - 1 m = 27 × 10 4 V Hence, the electric potential at the given point is 27 × 10 4 V .

Derive an expression for the energy stored in a parallel plate capacitor and express it in two more forms. The figure shows two air filled capacitors of capacitance 2 μF each connected in series. What will be the equivalent capacitance across A B if dielectric slabs of dielectric constant 3 are introduced between the plates of both the capacitors filling the space between the plates completely ?

Suppose, voltage V is applied across the capacitor of capacitance C and current I is flowing through it. Then, the power across the capacitor, P = V I Then P = V d q d t ....................... 1 The charge in capacitor, q = C V ⇒ V = q C ....................... 2 Substituting 2 in 1 we get: P = q C d q d t ⇒ P d t = q C d q ....................... 3 Now integrating both sides we get energy stored in the capacitor: U = q 2 2 C ....................... 4 The Electric field inside the capacitor: E = σ ε ∘ = q A ε ∘ ....................... 5 Here, σ is surface charge density of the capacitor plate, A and d ae the cross-sectional area and the the separation between the two capacitive plates. Since, the capacitance of the capacitor is C = ε ∘ A d ⇒ C d = ε ∘ A ....................... 6 Substituting equation 6 in equation 5 we get: E = q C d ⇒ q = E C d ....................... 7 Substituting equation 7 in equation 4 we get: U = E 2 C 2 d 2 2 C = E 2 C d 2 2 ....................... 8 Substituting C = ε ∘ A d in equation 8 we get: U = E 2 ε ∘ A d d 2 2 = U = E 2 ε ∘ A d 2 ....................... 9 Hence, the expressions for the parallel plate capacitor are: q 2 2 C E 2 C d 2 2 E 2 ε ∘ A d 2 Here both the capacitors are of same capacitance, i.e. C 1 = C 2 = 2 μF . To both of the capacitors dielectric slabs of dielectric constant 3 are introduced then the new capacitance of the capacitor is C 1 ' = C 2 ' = 3 × 2 μF = 6 μF . Then, the equivalent capacitance across: 1 C e q = 1 C 1 + 1 C 2 ⇒ C e q = C 1 C 2 C 1 + C 2 = 36 μF 2 12 μF = 3 μF Hence, the equivalent capacitance across A B for the given condition is 3 μF .

E M B I B E

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 2 - Electrostatic Potential and Capacitance>Goa Board-2019>Q 3

MEDIUM

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IMPORTANT

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A positive charge $+ q$ is located at a point P. What would be the value of the work done if another point charge $+ q$ , is taken from point A to point B on the circumference of a circle drawn about the point P ? Why?

Important Questions on Electrostatic Potential and Capacitance

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IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 2 - Electrostatic Potential and Capacitance>Goa Board-2019>Q 4

With the help of a diagram, obtain an expression for the effective capacitance of three capacitors connected in parallel. A voltage ‘V’ is applied across the combination. Also, write the general expression for the effective capacitance of n capacitors connected in parallel.

EASY

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IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR PHYSICS>Chapter 2 - Electrostatic Potential and Capacitance>Goa Board-2019>Q 5

What will be the effect on the voltage across each capacitor if one capacitor is removed from the above combination, which was three capacitors connected in parallel. A voltage ‘V’ is applied across the combination.