Direct Current

 CURRENT

Definitions:$\\$

• The flow of charge in a definite direction per second is called electric current.$\\$

• I=$\frac{q}{t}$=$\frac{ne}{t} \\$

• If I=1A,t=1 sec then,$\\$

• Number of electrons (n)=6.25×$10^18$ electron flowing in 1 A current per second$\\$

• current is a scalar Quantity.$\\$

Drift Velocity$\\$Drift velocity is defined as the average velocity with which free electrons get drifted towards the + ve end of the conductor under the influence of an external electric field.$\\$$\vec{V_d}$ =-$\frac{e\vec{E}}{m}$ τ$\\$τ → time of Relaxation.$\\$

|$\vec{V_d}$|= $10^{-4}$m/s, τ = $10^{-14}$ second.

Relaxation Time$\\$→ It is the average time that has elapsed since each electrons suffered its last collision with the ion or atom of the conductor, while drifting forwards the + ve end of the conductor under the effect of electric field applied.$\\$

mathematically,$\\$τ= $\frac{mean free path(λ)}{r⋅m⋅s velocity of electrons (v_{rms})} \\$Note:$\\$(1) τ∝$\frac{a}{temperature}$$\\$(ii) τ ∝ $\frac{1}{R} \\$

Current Density$\\$→ current density at a point inside the conductor is the amount of current flowing per unit area around that point of the conductor, provided the area is held in a direction normal to the current.$\\$

Current Density(J) = $\frac{I}{A} \\$SI unit of current density is A/$m^2 \\$It is a vector quantity. Its direction is that of the flow of +ve charge at the given point inside the conductor.$\\$I= JAcosθ$\\$=$\vec{J}.\vec{A} \\$$\vec{A}$ is the area vector of the plane.

Resistance

Basic Info$\\$Resistance is the obstruction possessed by the conductor to the flow of current through it.$\\$R∝ρ$\frac{l}{A} \\$ρ→ resistivity$\\$l → Length of conductor$\\$A→ Cross section Area of conductor$\\$And,$\\$ρ=$\frac{m}{ne^2τ} \\$P.S.

• Resistance depends upon: temperature, nature & dimension of material.$\\$

but,$\\$

• Resistivity depends only upon temperature and nature of material.$\\$

Ohm's Law$\\$It states that,"if the physical conditions such as temperature, nature and dimensions of conductor remains constant, the current between two points in a conductor is directly proportional to the potential difference between these two points.$\\$i.e,$\\$I ∝ R$\\$I=$\frac{1}{R}$V$\\$∴ V=IR$\\$In following graph I vs V, a straight line passing through origin is observed:$\\$

Note:$\\$

• A conductor which obeys ohm's law are called ohmic conductor.$\\$

E.g:- metals, alloys$\\$

• Conductors which do not obey ohm's law are called non-ohmic conductors.$\\$

E.g:- diode valve, neon gas , junction diode, carbon compounds etc.$\\$→ I vs V graph is parabola for Zener diode.$\\$

Resistance and Temperature:$\\$$R_t=R_o(1+α∆θ) \\$$R_o$ → resistance of the conductor at $0^o$C$\\$α→ temperature co-efficient$\\$∆θ→change in temperature$\\$∆θ=t-0= $t^oC \\$$R_t$→ resistance at $t^oC \\$

• For metals, α is positive.$\\$
• For semi-conductor and insulator, α is negative.$\\$
• Alloy's (eg:-maganin, eureka, constantan etc), have less value of α$\\$
• α= 0 for super conductors.$\\$

We can say, Resistance of super conductor is also 0.

Combination of Resistor

Series Combination$\\$

Key Points:$\\$a. $R_s$= $R_1 + R_2 + R_3 \\$b. Current through each resistor is same [Resistance]$\\$c. $R_s$ > $R_1 , R_2 , R_3$ (always)$\\$d. V across $R_1 , R_2 , R_3$ is different.$\\$e. $V_s$= $V_1 + V_2 + V_3 \\$f. $V_1 : V_2 : V_3$...... = $R_1 : R_2 : R_3$........$\\$g. Used in resistance box and decorative bulbs.$\\$

Parallel Combination$\\$

Key Points:$\\$a. $\frac{1}{R_{p}} = \frac{1}{R_{1}} + \frac{1}{R_{2}}+ \frac{1}{R_{3}}$ b. I = $I_1 + I_2 + I_3$
c. P.d across each resistance is same. d. $R_s$ < $R_1 , R_2 , R_3$ (always)$\\$e. $I_1 : I_2 : I_3$ ........ = $\frac{1}{R_{1}} + \frac{1}{R_{2}}+ \frac{1}{R_{3}}$ f. $\frac{R_s}{R_p}$ = $n^2$ g. Used in wiring h. Domestic Appliance

Electrical Cell, Internal Resistance, & Potential Difference

Electric cell is a device or apparatus which facilitates the motion of charges across the circuit.

The agent which facilities the motion of charges the circuit.

The agent which facilitates this motion of charges is Emf.

Emf(electromotive force) is not actually a force. It is work done$\\$facilitating the motion of charges.$\\$Emf depends upon:$\\$(i) the nature of two plates$\\$(ii) nature, temperature & concentration of electrolyte.$\\$

The internal resistance (r) is the resistance of the column of liquid$\\$between two plates of the cell.$\\$Depends Upon:$\\$(i) Seperation/ distance between the plates$\\$(ii) Area of cross section of the column$\\$(iii) Nature, temperature & degree of dissociation of electrolyte$\\$

Terminal potential difference between the two electrodes of a cell in a closed circuit (i.e when a curremt is drawn from the cell).

Playing with Cells

**Series Combination **$\\$

Let n identical cells are connected in series as shown.$\\$Then,$\\$Total emf = E+E+E+E.......................= nE$\\$

Total internal resistance = r+ r+ r+ r+.................= nr$\\$Total resistance = R+nr$\\$$\therefore$Current = $\frac{Total emf}{Total resistance} = \frac{nE}{R+nr} \\$

Parallel Combination$\\$

total current ; $I$= $I_1 + I_1+I_1+$............$\\$$I$= n$I_1 \\$$I_1$=$\frac{I}{n} \\$[current through individual cells]$\\$Total emf of cells = E [due to parallel combination]$\\$

Total resistance in the cicuit = R + $\frac{r}{n}$ [r is internal resistance across n columns]$\\$$I$ = $\frac{Total emf }{Total resistance}=\frac{E}{R+\frac{r}{n}} \\$

Special Condition$\\$(i) If R=0(i.e load is short circuited)$\\$$I$=$\frac{nE}{r}$ => I∝ n$\\$

(ii) If R>>$\frac{r}{n}$ (i.e nR>>r) then$\\$$I$ = $I_{min}$ =$\frac{E}{R}$ i.e current due to single cell.$\\$

(iii) If R>>$\frac{r}{n}$ (i.e nR>>r) then$\\$

$I$ = $I_{max}$ =$\frac{nE}{R} \\$

(iv) R=r then,$\\$

$I$ = $\frac{nE}{R(n+1)}$ = $\frac{nE}{r(n+1)} \\$

(v) V= E-$I\frac{r}{n}$, V-> Terminal potential difference of parallel combination$\\$

Mixed Combination*$\\$

Let cells each of emf E & internal resistance r be connected in m rows with n cells in each rows as shown in figure above, Total number of cells = nm Total emf = nE Total internal resistance = $\farc{nr}{m}$ Total Resistance =R+$\frac{nr}{m}$

I = $\frac{Total emf }{Total resistance}=\frac{nE}{R+\frac{nr}{m}} = \frac{mnE}{mR+nr} \\$Condition:$\\$

(i) For $I_{max} \\$mR=nr$\\$proof:$\\$$I_{max}$=$\frac{mnE}{mR+nr}$=$\frac{mnE}{2nr}$=$\frac{mE}{2r} \\$Also,$\\$$I_{max}$=$\frac{mnE}{mR+nr}$=$\frac{mnE}{2mR}$=$\frac{nE}{2R} \\$

P.S This combination is used when more power is required in the circuit.

WheatStone's Bridge

Description$\\$

If four resistances P,Q,R,S are arranged to structure as shown above, & if the galvanometer {G} shows no deflection (i.e. I along BC=0), then the bridge is balanced.Such, condition is called balanced wheatstone bridge.$\\$$\frac{P}{Q}=\frac{R}{S}$$\\$Key Points$\\$Meter bridge (slide wire bridge ), post office box are practical application of wheat stone bridge.

Potentiometer

Game changer device in electrical world.

A potentiometer consists of a long uniform wire generally made of manganin or constantan, stretched on a wooden board.

Principle$\\$It is based on the fact that the fall of potential across any portion of the wire is directly proportional to the length of that portion provided the wire is of uniform area of cross-section and constant current is flowing through it. i.e.$\\$V ∝ l$\\$[If I & A are constant]$\\$V = kl$\\$k -> the potential gradient , i.e fall of potential per unit length of the given wire.$\\$

P.S. All gradient are vectors.$\\$

Application$\\$(i) Measuring internal resistance of cell.$\\$(ii) Comparison of emfs of two cells by using potentiometer.$\\$

Measurment of Internal Resistance:$\\$r= R($\frac{l_1-l_2}{l_2}$)$\\$or,$\\$r=R($\frac{E}{V}-1$)$\\$

Comparison of emfs of two cells:$\\$$\frac{E_1}{E_2}=\farc{l_1}{l_2}$

NOTE$\\$The sensitivity of potentiometer can be increased by decreasing the potential gradient (k),$\\$Steps to increase the sensitivity of potentiometer:$\\$(i) By increasing the length of potentiometer wire.$\\$(ii) By reducing the current in the potentiometer wire from main battery with the help of rheostat, if the potentiometer wire is of fixed length.$\\$

P.S.$\\$Potentiometer is based on null deflection method.