Here students can locate TS Inter 2nd Year Physics Notes 4th Lesson Electric Charges and Fields to prepare for their exam.

## TS Inter 2nd Year Physics Notes 4th Lesson Electric Charges and Fields

→ Electrical charges given to conductors will flow from one end to other end. Charges moving through conductors leads to flow of current.

→ Electrical charges are two types

- positive charge,
- negative charge.

→ In the process of electrification we will remove or add electrons to substances with some techniques. Substance that looses electrons will become positive substance that gains electrons will become negative.

→ Quantisation of charge: Electric charge ‘Q ’ on a substance is an integral multiple of fundamental charge of electron, i.e., Q = ne. It is called Quantisation of charge.

→ Law of conservation of charge : The total charge of an isolated system is always con-stant. i.e., charge can not be created or des-troyed. This is known as “law of conservation of charge”.

→ Charge on electron e = 1.602 × 10^{-19} C it is taken as fundamental charge.

→ Coulomb’s Law:

Force attraction (or) repulsion between the charges is proportional to product of charges and inversely proportional to the square of the distance between them.

∴ From Coulomb’s law F ∝ q_{1}q_{2} and F ∝ 1/r^{2}

F ∝ \(\frac{q_1 q_2}{r^2}\) (OR) F = \(\frac{1}{4 \pi \varepsilon_0} \frac{q_1 q_2}{r^2}\)

Where \(\frac{1}{4 \pi \varepsilon_0}\) = 9 × 10^{9} N-m^{2} / C^{2} is a constant.

ε_{0} is permittivity of free space.

Its value is 8.85 × 10^{-12} farad/metre.

→ Force on a given charge (q) due to multiple charges is the vector sum of all the forces acting on the given charge.

\(\overline{\mathrm{F}_{\mathrm{R}}}=\overline{\mathrm{F}}_1+\overline{\mathrm{F}}_2+\overline{\mathrm{F}}_3\) ……….

Where F_{1} = \(\frac{1}{4 \pi \varepsilon_0} \frac{q^q q_2}{r_1^2}\) etc

Note : To find resultant force we must use principles of vector addition i.e., parallelogram law or triangle law.

→ Electric field : Every charged particle (q) will produce an electric field everywhere in the surrounding. It is a vector. It follows inverse square law.

→ Intensity of electric field (or) electric field strength (E̅) : Intensity of electric field or electric field at a point in space is the force experienced by a unit positive charge placed at that point.

F = Eq (or) E = (F/q), SI unit: Vm^{-1}

→ Electric field lines of force : Electric field lines represent electric field E due to a charge ‘q’ in a pictorial manner. When E is strong field lines are move nearer or crowded. In a weak field electric field lines are less dense.

→ Electric flux (Φ): The number of electric field line passing through unit area placed normal to the field at a given point is called “electric flux”. It is a measure for the strength of electric field at that point.

→ Electric dipole: Two equal and opposite charges separated by some distance will constitute an “electric dipole”.

→ Dipole moment (p̅): The product of one of the charge in dipole and separation between the charges is defined as “dipole moment (P̅)”.

Dipole moment (p̅) = q. 2a (or) p = 2aq

It is a vector. It acts along the direction of -q to q.

Unit: coulomb – metre.

→ Dipole in a uniform electric field : Let an electric dipole is placed in an electric field of intensity E. Then F = Eq

Torque on dipole τ = p̅ x E̅

Let p̅ and E̅ are in the plane of the paper then torque τ will act perpendicularly to the plane of the paper.

→ Linear charge density (λ) :

It is defined as the ratio of charge (Q) to length of the conductor (L).

Linear charge density

λ = \(\frac{\text { Charge }}{\text { Length }} \frac{(\mathrm{Q})}{(\mathrm{L})}\)

⇒ λ = \(\frac{\Delta \mathrm{Q}}{\Delta \mathrm{L}}\)

Unit: Coulomb / metre.

→ Surface charge density (σ) : W

It is defined as the ratio of charge (Q) to surface area of (A) of that conductor.

Surface charge density

σ = \(\frac{\text { Charge }}{\text { Area }} \frac{(\mathrm{Q})}{(\mathrm{A})}\)

⇒ σ = \(\frac{\Delta \mathrm{Q}}{\Delta \mathrm{A}}\)

Unit: Coulomb / metre^{2}

→ Volume charge density (ρ):

It is defined as the ratio of charge on the conductor ‘Q’ to volume of the conductor.

Volume charge density

ρ = \(\frac{\text { Charge }}{\text { Volume }} \frac{(Q)}{(V)}\)

⇒ ρ = \(\frac{\Delta \mathrm{Q}}{\Delta \mathrm{V}}\)

Unit: Coulomb / m^{3}.

→ Gauss law : The total electrical flux (Φ) through a closed surface (s) is 1/ε_{0} times more than the total charge (Q) enclosed by that surface.

From Gauss law (Φ) = \(\frac{1}{\varepsilon_0}\) Q

→ Important conclusions from Gauss’s law:

- Gauss law is applicable to any closed surface irrespective of its shape.
- The term Q refers to sum of all the charges inside the gaussian surface.
- A gaussian surface is that surface for which we choosed to apply gauss law.
- It is not necessary to consider any charges out side the gaussian surface to find the flux (Φ) coming out of it.
- Gauss law is very useful in the calculations to find electric field when the system (gaussian surface) has some symmetry.

→ Charge Q = ne. Where e = charge on electron = 1.6 × 10^{-19} C.

→ Force between two charges F = \(\frac{1}{4 \pi \varepsilon_0} \frac{q_1 q_2}{r^2}\)

→ Force between multiple charges : In a system of charges say q_{1} q_{2}, q_{3} ………. q_{n}.

Force on charge qj is say F_{1} = F_{12} + F_{13} + F_{14} ………….. F_{1n}

(OR) Total force on q_{1} say

F_{1} = \(\frac{1}{4 \pi \varepsilon_0}\left[\frac{q_1 q_2}{r_{21}^2}+\frac{q_1 q_3}{r_{13}^2}+\ldots . .+\frac{q_1 q_n}{r_{1 n}^2}\right]\)

→ Electric field due to a point charge q’ at a point r is E = \(\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{Q}}{\mathrm{r}^2}\) (OR) E̅ = \(\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{Q}}{\mathrm{r}^2}\)r̅

→ Electric flux Φ = E.Δs = E Δs cos θ.

Where θ is the angle between electric field E̅ and area vector Δs .

→ Dipole moment p = q.2a. Where q is one of the charge on dipole and ‘2a’ is separation between the charges.

→ Electric field at any point on the axis of a dipole

E_{axial} = \(\frac{\mathrm{q}}{4 \pi \varepsilon_0} \frac{4 \mathrm{ar}}{\left[\mathrm{r}^2-\mathrm{a}^2\right]^2}=\frac{1}{4 \pi \varepsilon_0} \frac{2 \mathrm{pr}}{\left(\mathrm{r}^2-\mathrm{a}^2\right)^2}\)

where r > > a then E_{axial} = \(\frac{1}{4 \pi \varepsilon_0} \frac{2 \mathrm{p}}{\mathrm{r}^3}\)

When r is the distance of given point from centre of dipole.

→ Electric field of any point on the equatorial line of a dipole.

E_{eq} = \(\frac{1}{4 \pi \varepsilon_0} \frac{2 \mathrm{qa}}{\left[\mathrm{r}^2+\mathrm{a}^2\right]^{3 / 2}}=\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{p}}{\left[\mathrm{r}^2+\mathrm{a}^2\right]^{3 / 2}}\)

When r >> a then E_{eq} = \(\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{p}}{\mathrm{r}^3}\)

Note : E_{axial} and E_{eq} will act along the line joining the given point ’p’ and centre of dipole ‘O’.

→ Torque on a dipole when placed in a uniform electric field E is τ̅ = p̅ x E̅ = pE sin θ.

Where θ is the angle between P̅ and E̅.

→ Charge distribution on conductors:

Charge (Q) given to a conductor will uniformly spread on the entire conductor.

(a) Linear charge density λ = \(\frac{\text { Charge }}{\text { Length }} \frac{\mathrm{Q}}{\mathrm{L}}\) unit: C/m

(b) Surface charge density σ = \(\frac{\text { Charge }}{\text { Surface area }} \frac{Q}{A}\) unit / C/m^{2}

(c) Volume charge density ρ = \(\frac{\text { Charge }}{\text { Volume }} \frac{Q}{V}\) unit: C/m^{3}

→ Gauss’s law : The total electric flux (Φ) coming out of a closed surface is \(\frac{1}{\varepsilon_0}\) times greater than the total charge (Q) enclosed by that surface.

Φ = \(\frac{Q}{\varepsilon_0}\)

→ Electric field due to an infinitely long straight uniformly charged wire E = \(\frac{\lambda}{2 \pi \varepsilon_0 r}\)n̅.

(∵ n̅ = 1) or, E = \(\frac{\lambda}{2 \pi \varepsilon_0 r}\)

→ Field due to a uniformly charged infinite plane sheet is E = \(\frac{\sigma}{2 \varepsilon_0}\)n̅.

or. E = \(\frac{\sigma}{2 \varepsilon_0}\), (∵ n̅ = 1)

→ Field due to a uniformly charged thin spherical shell:

(a) At any point out side the shell is

E = \(\frac{\mathrm{Q}}{4 \pi \varepsilon_0 \mathrm{r}^2}\)

(b) Inside the shell electric field E = 0.