Here students can locate TS Inter 2nd Year Physics Notes 6th Lesson Current Electricity to prepare for their exam.

## TS Inter 2nd Year Physics Notes 6th Lesson Current Electricity

→ Ohm’s Law: At constant temperature current (I) flowing through a conductor is proportional to the potential difference between the ends of that conductor.

V ∝ I ⇒ V = RI where R = constant called resistance. Unit: Ohm (Ω).

→ Conductors – Resistance:

Resistance: The obstruction created by a conductor for the mobility of charges through it is known as “resistance”.

- The resistance of a conductor (R) is proportional to length; R ∝ l → (1)
- and Inversely proportional to area of cross section of the conductor.

R ∝ l → (2)

From eq. (1) & (2) R ∝ \(\frac{l}{\mathrm{~A}}\) ⇒ R = \(\frac{\rho l}{\mathrm{~A}}\)

⇒ ρ = \(\frac{\mathrm{RA}}{l}\)

where p = resistivity of the conductor.

→ Current density (J):

The ratio of current through a conductor to its area of cross-sec¬tion is called “current density (j).”

Current density (j) = \(\frac{\text { Current }}{\text { Area }}=\frac{I}{A}\)

Note:

- Potential V = IR = I\(\frac{\rho \cdot l}{\mathrm{~A}}\) = jρl
- Potential V = E.I. (Intensity of electric field x distance)

∴ EI = JpI or E = jp or j = \(\frac{E}{\rho}\) = Eσ

where σ Is conductivity of the material.

→ Drift velocity (v_{d}): The speed with which an electron gets drifted in a metallic conductor under the application of external electric field is called “drift velocity (v_{d}).”

→ Drift Velocity v_{d} = \(\frac{-\mathrm{eE}}{\mathrm{m}}\)τ. Where τ = the average time between two successive collisions.

Note: Current density j = \(\frac{\mathrm{ne}^2}{\mathrm{~m}}\)τE and Conductivity = σ = \(\frac{\mathrm{ne}^2}{\mathrm{~m}}\)τ.

→ Mobility (μ): It is defined as the mag-nitude of drift velocity per unit electric field.

μ = \(\frac{\left|\mathrm{v}_{\mathrm{d}}\right|}{\mathrm{E}}\) But vd = \(\frac{\mathrm{e} \mathrm{E}}{\mathrm{m}}\) ⇒ μ = \(\frac{v_d}{E}=\frac{e \tau}{m}\)

→ Resistivity: Resistivity of a substance

ρ = \(\frac{\mathrm{RA}}{\ell}\)

It is defined as the resistance of a unit cube between its opposite parallel surfaces.

Resistivity depends on the nature of substance but not on its dimensions.

Unit: Ohm – metre (Ωm).

→ Temperature coefficient of resistivity:

The resistivity of a substance changes with temperature. ρ_{T} = ρ_{0} [l + α(T – T_{0})].

Where α = temperature coefficient of resistivity.

α = \(\)/°C

Note:

- For metals a Increases with temperature.
- For semiconductors and insulators a decreases with temperature.

→ Colour code: Carbon resistors have a set of coaxial coloured rings on them. It gives the value of that resistor along with tole¬rable limit.

On every carbon resistor four colour bands are printed. In some cases only Three colour bands are printed.

1st two bands from left to right gives the numerical values of that resistor.

3rd band gives number of zero’s to be put after first two digits.

Fourth band gives maximum allowed variation limit of that resistor called “tolerance”.

→ Colour code – Values:

- Black → 0;
- Brown → 1;
- Red → 2
- Orange → 3;
- Yellow → 4
- Green → 5
- Blue → 6;
- Violet → 7;
- Gray → 8;
- White → 9

→ Tolerance: Gold band – 5 %; Silver band -10 %. If there is no 4th band tolerance is 20%. Ex: A carbon resistor consists of Orange, green, green bands then its value is

1st orange = 3. 2nd band green = 5,

3rd green = 5

No Fourth band ⇒ tolerance is 20%

So for that resistor the value is 35 followed by five zero’s.

∴ Resistance of resistor is 3500000 Q

i. e., R = 3.5 Mega ohms with 20% tolerance.

→ Electrical Power (P): Energy dissipated per unit time is “power”.

In a conductor of resistance R’ while carrying a current I, this power produces heat in that conductor.

Power P = I^{2}R = VI = V^{2}/R. Unit: Watt.

→ Transmission power loss (P_{c}): Power wasted in transmitting lines P . While supplying electrical power from generator to consumer is P_{c} = I^{2}R_{c} => P_{c} = \(\frac{\mathrm{P}^2 \mathrm{R}_{\mathrm{c}}}{\mathrm{V}^2}\)

i. e., power wasted in a line is inversely proportional to the square of voltage of line. P = total power to be transmitted.

∵ P_{c} ∝ \(\frac{1}{\mathrm{~V}^2}\) we are prefering high voltage transmission lines to reduce transmission power losses.

→ Resistors in series: When resistances are connected in series (1) Same current flows through all resistors.

Effective resistance (R_{eff}) is the sum of individual resistances i.e., R_{eff} = R_{1} + R_{2} + R_{3} + ……………..

ii) Effective resistance R_{eff} is greater than the greatest value of resistor in that combination.

→ Resistors in parallel:

- When resistors are connected in parallel potential difference across all resistors is same.
- Effective resistance is given by \(\frac{1}{R_{\text {efl }}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\)
- Effective resistance is less than least resistor in that combination.

→ Cells emf and Internal resistance: emf of a cell (ε):

The open circuit voltage between negative and positive terminals of a cell is called “emf of that cell”.

→ Internal resistance of cell (r):

In a cell current flows through electrolyte. Every electrolyte has some finite resistance.

The resistance offered by the cell for the flow of current through it is called “internal resistance of the cell (r).

Note: When a cell is connected in a circuit then potential difference across terminals is V = ε – ir

Current in the circuit i = ε/(R + r)

Where E = emf of cell, r = internal resis-tance and R = resistance of the circuit.

→ Cells in series:

Let two cells of emf ε_{1} and ε_{2} with internal resistance r_{1}, and r_{2} are connected in series then

i. e., power wasted in a line is inversely proportional to the square of voltage of line. P = total power to be transmitted.

∵ P_{c} ∝ \(\frac{1}{\mathrm{~V}^2}\) we are prefering high voltage transmission lines to reduce transmission power losses.

→ Resistors in series: When resistances are connected in series (1) Same current flows through all resistors.

Effective resistance (R_{eff}) is the sum of individual resistances i.e., R_{eff} = R_{1} + R_{2} + R_{3} + …………

ii) Effective resistance R_{eff} is greater than the greatest value of resistor in that combination.

→ Resistors in parallel:

- When resistors are connected in parallel potential difference across all resistors is same.
- Effective resistance is given by

\(\frac{1}{\mathrm{R}_{\mathrm{efl}}}=\frac{1}{\mathrm{R}_1}+\frac{1}{\mathrm{R}_2}+\frac{1}{\mathrm{R}_3}\) - Effective resistance is less than least resistor in that combination.

→ Cells emf and Internal resistance : emf of a cell (ε) : The open circuit voltage between negative and positive terminals of a cell is called “emf of that cell”.

→ Internal resistance of cell (r) : In a cell current flows through electrolyte. Every electrolyte has some finite resistance.

The resistance offered by the cell for the flow of current through it is called “internal resistance of the cell (r).”

Note: When a cell is connected in a circuit then potential difference across terminals is V = ε – ir

Current in the circuit i = ε/(R + r)

Where E = emf of cell,

r = internal resistance and

R = resistance of the circuit.

→ Cells in series: Let two cells of emf ε_{1} and ε_{2} with internal resistance r_{1}, and r_{2} are

- total emf ε = ε
_{1}+ ε_{2} - total p.d across them

V = ε_{1}+ ε_{2}– i(r_{1}+ r_{2}) - equivalent resistance r
_{eq}= r_{1}+ r_{2} - current in circuit I = \(\frac{\varepsilon_1+\varepsilon_2}{R+r_1+r_2}\)
- Due to series combination potential difference in the circuit increases.

Note : If n identical cells are connected in series emf ε = nε_{1}, I = nI_{1}, r_{eff} = n .r .

Where ε_{1} = emf of single cell,

I_{1} = current given by single cell in circuit

r = internal resistance of each cell.

→ Parallel combination of cells: Let two cells of emf ε_{1} and ε_{2} are connected parallel then

- Equivalent emf ε
_{eq}= \(\frac{\varepsilon_1 \mathbf{r}_2+\varepsilon_2 r_1}{r_1+r_2}\) - Equivalent resistance \(\left(\frac{1}{r_{\mathrm{eq}}}\right)=\frac{1}{r_1}+\frac{1}{r_2}\)

⇒ r_{eq}= \(\frac{r_1 r_2}{r_1+r_2}\) - Current in circuit I = I
_{1}+ I_{2} - P.D in circuit V = \(\frac{\varepsilon_1 r_2+\varepsilon_2 r_1}{r_1+r_2}-I\left[\frac{r_1 r_2}{r_1+r_2}\right]\)

⇒ V_{eq}= ε_{eq}– Ir_{eq} - Due to parallel combination current in the circuit increases.

Note: When n identical batteries are connected in parallel.

- emf ε
_{1}= ε - Current I = nI
_{1} - effective internal resistance of combination r
_{eff}= \(\frac{r}{n}\)

→ Kirchhoff s Laws:

- Junction rule : At any junction sum of currents towards the junction is equals to sum of currents away from the junction. (OR) Algebraic sum of currents around a junction is zero.
- Loop rule : Algebraic sum of changes in potential around any closed loop involving resistors and cells in the loop is zero.

→ Wheatstone’s principle : In a balanced Wheatstone’s bridge ratio of resistances in adjacent arms is constant.

i.e \(\frac{P}{Q}=\frac{R}{S}\Rightarrow \frac{\mathrm{R}_1}{\mathrm{R}_2}=\frac{\mathrm{R}_3}{\mathrm{R}_4}\)

⇒ \(\frac{\mathrm{P}}{\mathrm{Q}}=\frac{l}{(100-l)}\)

→ Average current I = \(\frac{\Delta q}{\Delta t}\); Instantaneous current i = \(\frac{\mathrm{dq}}{\mathrm{dt}}\); Current density j = i/A Unit : Amp/rn2

→ Resistance R = \(\frac{V}{i}\); Resistance R = \(\frac{\rho l}{\mathrm{~A}}=\frac{\rho l}{\pi \mathrm{r}^2}\); Conductance G = \(\frac{1}{R}\).

→ AcceleratIon of electron In electric field a = \(\frac{\mathrm{eE}}{\mathrm{m}}\)

→ Drift velocity of electron (v_{d}) = \(\frac{i}{\text { neA }}\), V_{d} = \(\frac{e \tau E}{m}\); mobility (μ) = \(\frac{\mathrm{e \tau}}{\mathrm{m}}\) where τ average time between two successive collisions.

→ ResIstivity (or) specific resistance

ρ = \(\frac{\mathrm{RA}}{l}=\frac{\mathrm{R} \pi \mathrm{r}^2}{l}\); Conductance σ = \(\frac{1}{\rho}\)

→ Temperature coefficient of resistivity

α = \(\frac{\rho_2-\rho_1}{\rho_1\left(t_2-t_1\right)}\)/°C ⇒ α = \(\frac{\mathrm{d} \rho}{\rho \mathrm{dt}}\)/°C

→ Temperature coefficient of resistance

α = \(\frac{\mathrm{R}_{\mathrm{t}}-\mathrm{R}_0}{\mathrm{R}_0\left(\mathrm{t}_2-\mathrm{t}_1\right)}\)/°C

α = \(\frac{\mathrm{dR}}{\mathrm{Rdt}}\) (or) α = \(\frac{\mathrm{R}_2-\mathrm{R}_1}{\mathrm{R}_1\left(\mathrm{t}_2-\mathrm{t}_1\right)}\)/°C

R_{t} = R_{0}[1 + α(t_{2} – t_{1})]

→ If two wires are made of same material then

\(\frac{\mathrm{R}_1}{\mathrm{R}_2}=\frac{l_1}{l_2} \frac{\mathrm{A}_2}{\mathrm{~A}_1} \Rightarrow \frac{\mathrm{R}_1}{\mathrm{R}_2}=\frac{l_1 \mathrm{r}_2^2}{l_2 \mathrm{r}_1^2}\)

→ In series combination of resistors:

- R
_{eq}= R_{1}+ R_{2}+ …………. + R_{n} - Same current will flow through all resistors.
- Potential drop on resistors V
_{1}= i R_{1}, V_{2}= i R_{2}etc. - Total potential drop V = V
_{1}+ V_{2}+ ……………. etc. - Current of circuit I = \(\frac{\text { Total Potential }}{\text { Total Resistance }}=\frac{V}{R_{e q}}\)

→ In parallel combination of resistors:

- \(\frac{1}{R_{e q}}=\frac{1}{R_1}+\frac{1}{R_2}+\ldots \ldots \ldots+\frac{1}{R_n}\)
- For two resistors (R
_{eg}) = \(\frac{\mathrm{R}_1 \mathrm{R}_2}{\mathrm{R}_1+\mathrm{R}_2}\) - Same potential difference is applied on all resistors.
- Total current I = I
_{1}+ I_{2}+ I_{3}+ ……… i.e., sum of individual currents through each resistor.

I = \(\frac{\mathrm{V}}{\mathrm{R}_{\mathrm{eq}}}=\mathrm{V}\left(\frac{1}{\mathrm{R}_1}+\frac{1}{\mathrm{R}_2}+\ldots .+\frac{1}{\mathrm{R}_{\mathrm{n}}}\right)\)

→ In cells

- While discharging, termina! voltage V = E – ir
- While charging. terminal voltage V = E + ir
- Current in circuIt i = \(\frac{E}{R+r}\)

r = Internal resistance of battery;

R = Resistance in circuit.

→ Electrical energy W = Vit = i^{2}Rt = \(\frac{\mathrm{V}^2}{\mathrm{R}}\)t

→ Electric power P = Vi = i^{2}R = \(\frac{\mathrm{V}^2}{\mathrm{R}}\)t; Power wasted in transmission lines P_{c} = P^{2}R_{c}/V^{2} where P = Power transmitted ;

R_{c} = Resistance of line

→ 1 kilo watt hour = 36 × 10^{5} J or 3.6 × 10^{6} J.

→ At balance condition in Wheatstone’s bridge \(\frac{P}{Q}=\frac{R}{S}\)

→ If capacitors are used in balanced Wheat stone’s bridge \(\frac{C_1}{C_2}=\frac{C_3}{C_4}\)

→ In meter bridge at balance condition \(\frac{\mathrm{R}}{\mathrm{S}}=\frac{l_1}{l_2}\)

Unknown resistance x = R\(\frac{l_1}{l_2}\)

where I_{2} = (100 – I_{1})

→ In potentiometer,

In comparison of emf of two cells \(\frac{\mathrm{E}_1}{\mathrm{E}_2}=\frac{l_1}{l_2}\)

→ In determination of internal resistance

r = R\(\left[\frac{l_1-l_2}{l_2}\right]\)