Physics Standard 12 Vol I>Chapter -1 - Electrostatics>EVALUATION>Q 10

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12th Tamil Nadu Board

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A thin conducting spherical shell of radius $R$ has a charge $Q$ which is uniformly distributed on its surface. The correct plot for electrostatic potential due to this spherical shell is

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Important Points to Remember in Chapter -1 - Electrostatics from Tamil Nadu Board Physics Standard 12 Vol I Solutions

1. Basic properties of electric charges:

(i) There are two types of charges in nature; positive charge and negative charge. Like charges repel each-other and unlike charges attract each-other.

(ii) Conductors allow movement of electric charge through them, insulators do not.

(iii) Electric charge has three basic properties: quantisation, additivity and conservation.

(iv) Quantisation of electric charge means that total charge $(q)$ of a body is always an integral multiple of a basic quantum of charge $(e)$ i.e., $q = \pm n e$

(v) Additivity of electric charges means that the total charge of a system is the algebraic sum of all individual charges in the system.

(vi) Conservation of electric charges means that the total charge of an isolated system remains unchanged with time.

2. Coulomb's law and superposition principle:

(i) Coulomb’s Law: The mutual electrostatic force between two point charges $q_{1}$ and $q_{2}$ is proportional to the product $q_{1} q_{2}$ and inversely proportional to the square of the distance $r_{21}$ separating them.

(ii) Mathematically coulomb's law is given as: $F_{21} =$ force on $q_{2}$ due to $q_{1} = \frac{k (q_{1} q_{2})}{r_{21}^{2}} {\hat{r}}_{21}$ where ${\hat{r}}_{21}$ is a unit vector in the direction from $q_{1}$ to $q_{2}$ and $k = \frac{1}{4 π ε_{0}}$ is the constant of proportionality. $ε_{0} = 8.854 \times 10^{- 12} C^{2} N^{- 1} m^{- 2}$ and $k = 9 \times 10^{9} N m^{2} C^{- 2}$

(ii) Superposition Principle: For an assembly of charges $q_{1}, q_{2}, q_{3}, \dots$ the force on any charge, say $q_{1}$ , is the vector sum of the force on $q_{1}$ due to $q_{2}$ due to $q_{3}$ , and so on. For each pair, the force is given by the Coulomb’s law for two charges.

3. Electric field:

(i) The space around a charge in which its electrical effects can be observed is called electric field of that charge.

(ii) Electric field due to a point charge $q$ has a magnitude $| q | / 4 π ε_{0} r^{2}$ . it is radially outwards from $q$ , if $q$ is positive, and radially inwards if $q$ is negative. Like Coulomb force, electric field also satisfies superposition principle.

4. Electric field lines:

(i) An electric field line is a curve drawn in such a way that the tangent at each point on the curve gives the direction of electric field at that point.

(ii) Field lines are continuous curves without any breaks.

(iii) Two field lines cannot cross each other.

(iv) Electrostatic field lines start at positive charges and end at negative charges.

5. Electric dipole:

(i) An electric dipole is a pair of equal and opposite charges $q$ and $- q$ separated by some distance $2 a$ . Its dipole moment vector $p$ has magnitude $2 q a$ and is in the direction of the dipole axis from $- q$ to $q$ .

(ii) Field of an electric dipole in its equatorial plane at a distance $r$ from the centre: $E = \frac{- p}{4 π ε_{o}} \frac{1}{{(a^{2} + r^{2})}^{3 / 2}}$ $≅ \frac{- p}{4 π ε_{o} r^{3}}$ for $r ≫ a$

(iii) Dipole electric field on the axis at a distance $r$ from the centre: $E = \frac{2 p r}{4 π ε_{0} {(r^{2} - a^{2})}^{2}}$ $≅ \frac{2 p}{4 π ε_{0} r^{3}}$ for $r ≫ a$

(iv) In a uniform electric field $E$ , a dipole experiences a torque $τ$ given by, $τ = p \times E$ but experiences no net force.

6. Gauss' law:

(i) The flux $Δ ϕ$ of electric field $E$ through a small area element $Δ S$ is given by: $Δ ϕ = E \cdot Δ S$ where $Δ S$ is the area vector to the surface and it is taken positive along the outward normal to the surface.

(ii) Gauss’ law: The flux of electric field through any closed surface $S$ is $1 / ε_{0}$ times the total charge enclosed by $S$ .

7. Application of gauss' law:

(i) Electric field due to the thin infinitely long straight wire of uniform linear charge density $λ$ , $E = \frac{λ}{2 π ε_{0} r} \hat{n}$ where $r$ is the perpendicular distance of the point from the wire and $\hat{n}$ is the radial unit vector in the plane normal to the wire passing through the point.

(ii) Electric field due to infinite thin plane sheet of uniform surface charge density $σ$ : $E = \frac{σ}{2 ε_{0}} \hat{n}$ where $\hat{n}$ is a unit vector normal to the plane, outward on either side.

(iii) Electric field due to thin spherical shell of uniform surface charge density $σ$ :
$E = \frac{q}{4 π ε_{0} r^{2}} \hat{r}$ $(\begin{matrix} r \geq R \end{matrix})$
$E = 0$ $(r < R)$ where $r$ is the distance of the point from the centre of the shell and $R$ the radius of the shell. $q$ is the total charge of the shell.

8. Electrostatic potential:

(i) Electrostatic force is a conservative force. Work done by an external force in bringing a charge $q$ from a point $R$ to a point $P$ is $U_{P} - U_{R}$ which is the difference in potential energy of charge $q$ between the final and initial points.

(ii) Potential at a point is the work done per unit charge (by an external agency) in bringing a charge from infinity to that point. If potential at infinity is chosen to be zero, potential at a point with position vector $r$ due to a point charge $Q$ placed at the origin is given is given by, $V (r) = \frac{1}{4 π ε_{o}} \frac{Q}{r}$

(iii) The electrostatic potential at a point with position vector $r$ due to a point dipole of dipole moment $p$ placed at the origin is, $V (r) = \frac{1}{4 π ε_{o}} \frac{p . \hat{r}}{r^{2}}$

(iv) For a charge configuration $q_{1}, q_{2}, \dots, q_{n}$ with position vectors $r_{1}, r_{2}, \dots r_{n}$ the potential at a point $P$ is given by the superposition principle, $V = \frac{1}{4 π ε_{0}} (\frac{q_{1}}{r_{1 p}} + \frac{q_{2}}{r_{2 P}} + \dots + \frac{q_{n}}{r_{n P}})$ where $r_{1 P}$ is the distance between $q_{1}$ and $P$ , as and so on.

(v) An equipotential surface is a surface over which potential has a constant value. The electric field $E$ at a point is perpendicular to the equipotential surface through the point.

9. Electrostatic potential energy:

(i) Potential energy stored in a system of charges is the work done (by an external agency) in assembling the charges at their locations. Potential energy of two charges $q_{1}, q_{2}$ at $r_{1}, r_{2}$ given by, $U = \frac{1}{4 π ε_{0}} \frac{q_{1} q_{2}}{r_{12}}$ where $r_{12}$ is distance between $q_{1}$ and $q_{2}$ .

(ii) The potential energy of a charge $q$ in an external potential $V (r)$ is $q V (r)$ . The potential energy of a dipole moment $p$ in a uniform electric field $E$ is $- p \cdot E$

10. Conductor in electrostatic field:

Electrostatics field $E$ is zero in the interior of a conductor, just outside the surface of a charged conductor, $E$ is normal to the surface given by $E = \frac{σ}{ε_{0}} \hat{n}$ where $\hat{n}$ is the unit vector along the outward normal to the surface and $σ$ is the surface charge density.

11. Capacitor:

(i) A capacitor is a system of two conductors separated by an insulator. Its capacitance is defined by $C = Q / V$ where $Q$ and $- Q$ are the charges on the two conductors and $V$ is the potential difference between them.

(ii) For a parallel plate capacitor (with vacuum between the plates), $C = ε_{0} \frac{A}{d}$ where $A$ is the area of each plate and $d$ the separation between them.

(iii) If the medium between the plates of a capacitor is filled with an insulating substance (dielectric), the net electric field inside the dielectric and hence the potential difference between the plates is reduced. Consequently, the capacitance $C$ increases from its value $C_{0} .$ When there is no medium (vacuum), $C = K C_{0}$ where $K$ is the dielectric constant of the insulating substance.

12. Combination of capacitors:

(i) For capacitors in the series combination, the total capacitance $C$ is given by: $\frac{1}{C} = \frac{1}{C_{1}} + \frac{1}{C_{2}} + \frac{1}{C_{3}} + \dots$
where $C_{1}, C_{2}, C_{3} \dots$ are individual capacitances.

(ii) In the parallel combination, the total capacitance $C$ is: $C = C_{1} + C_{2} + C_{3} + \dots$
where $C_{1}, C_{2}, C_{3} \dots$ are individual capacitances.

13. Energy stored in capacitor:

(i) The energy $U$ stored in a capacitor of capacitance $C$ , with charge $Q$ and voltage $V$ is $U = \frac{1}{2} Q V = \frac{1}{2} C V^{2} = \frac{1}{2} \frac{Q^{2}}{C}$