DC Circuit

aDC CircuitElectric CurrentAn electric current is a flow of electric charge in a circuit. More specifically, the electric current is the rate of charge flow past a given point in an electric circuit. The charge can be negatively charged electrons or positive charge carriers including protons, positive ions or holes.I = q

t$$I=\frac{q}{t}$$ If small charge dq flows in time dt then, I= dq

dt$$I=\frac{dq}{dt}$$ SI unit of current is Ampere. 1 Ampere current is defined as the current flowing through a conductor when 1coulomb of charge flows through the conductor in 1 second. Drift velocity:There is large no. of free electron continuously move from lower potential to higher potential. On the way or during motion the electron suffer collision with the positive ions of the conductor at that situation electron looses some of the energy. As in that way there are no. of collision and they loose more energy and due to lose of the energy electron cannot gets it actual velocity the net effect is that the electron drift with small velocity called drift velocity.Draft velocity of electron:Metallic conductor there are large no. of free electron and these free electron are responding only for carrying electricity through the conductor. In the absence of the electric field the resultant velocity of the electron in a particular direction is zero and hence the net transport of charge is also zero and the current don’t flow through the conductor in the absence of electric field.But if the electric field is applied then the electron start to accelerate in the direction of the field applied so, the velocity of the electrons increase which implies that K.E. of the electron also increases. When these electrons collide with the atoms or ions then the velocity of electrons decrease by transformation of energy to the atoms or ions this process is continuous and the average acceleration of an electron is ceases i.e. reduced to zero. Then the electron required a constant velocity to the opposite direction of the field applied. This constant velocity of the electron on the opposite direction of the field is called drift velocity of the electron.Direction of CurrentThere is often a lot of misunderstanding about conventional current flow and electron flow. This can be a little confusing at first but it is really quite straightforward.The particles that carry charge along conductors are free electrons. The electric field direction within a circuit is by definition the direction that positive test charges are pushed. Thus, these negatively charged electrons move in the direction opposite the electric field.

Current DensityThe amount of electric current traveling per unit cross-section area is called as current density and expressed in amperes per square meter. More the current in a conductor, the higher will be the current density. It is denoted by J. J = I

A$$J=\frac{I}{A}$$It is vector quantity and direction is in the direction of current.Relation between the electric current and drift velocity:Let us consider a conductor having length L, cross-sectional area A, containing ‘n’ no. of electron per unit volume, having charge e in each electron.

Volume of the given conductor =AL Total no. of free electron in given conductor ( q)=nAL Total no. of the charge in given conductor= nALe If the external source is connected to the conductor then charge starts to flow through the conductor. Let Vd${\displaystyle V_d}$ be the drift velocity of the electron, q be the charge passes through the length of conductor L in the time t.I be the current passes through the conductor is given by I=q

t =neAL

t $$I=\frac{q}{t}=neA\frac{L}{t}$$ But, Vd = L

t is the drift velocity. So the current in the conductor is $$But,V_d=\frac{L}{t}isthedriftvelocity.Sothecurrentintheconductoris$$ I = n e A VdI = Vd e n A$$\begin{array}{c}I=neA{V}_d\\ I=V_{d}enA\end{array}$$This the the relation between current and drift velocity. Again, drift velocity is Vd= I

nAe$$V_d=\frac{I}{nAe}$$and current density is J = I

A$$J=\frac{I}{A}$$ So, J =n e Vd$$J=ne{V}_d$$ This is the expression forRelation between the electric current and drift velocity. Ohm's lawNote: The conductors which obey ohm’s law strictly are called Ohmic conductors. The conductors which do not follow ohm’s law are called non – ohmic conductors.It states that the electric current flowing through any conductor is directly proportional to the potential difference between the ends of the conductor, under constant physical condition (temperature, pressure, etc.) Let I be the electric current flowing through any conductor and V be the potential difference between the ends the conductors. So, according to Ohm’s law, we can write: V ∝ I.Or, V = RIWhere, R is proportionality constant called resistance of the conductor.Experimental Verifiration of Ohm's Law

The experimental arrangement for verification of Ohm’s law is shown in figure. A and V stands for Ammeter and Voltmeter. Xh for Rheostat. Here, by varying the Rh, the electric current (I) and potential difference across any length of conductor is measured from Ammeter and Voltmeter respectively. We plot V versus I. If the plot is a straight line passing through origin i.e. of the form y = mx, which means current is directly proportional to the voltage applied. Hence Ohm’s law is said to be verified. Electrical ResistanceResistance (also known as ohmic resistance or electrical resistance) is a measure of the opposition to current flow in an electrical circuit. Resistance is measured in ohms, symbolized by the Greek letter omega (Ω).It is measured by the ratio of the potential difference V across its ends to the current I flowing through it. R = V

I$$R=\frac{V}{I}$$1 Ohm= 1 volt

1 ampere${\displaystyle \frac{1volt}{1ampere}}$The resistance of a condutor is said to be one ohm if one ampere current flows through it under a potential difference of one volt. Cause of Electric ResistanceAn electric current flows when electrons move through a conductor, such as a metal wire. The moving electrons can collide with the ions in the metal. This makes it more difficult for the current to flow, and causes resistance.Resistance depends on the following factors: Length of the conductorArea of cross-sectionMaterial of constructionVariation of reistance with temperatureReisitance of a conductor generally increases with the increase in temperature. If a conductor has reisitance R${\displaystyle R}$ 1 at 𝜃${\displaystyle 𝜃}$10C and R${\displaystyle R}$2 at 𝜃${\displaystyle 𝜃}$20C , then the increase in resistance ( R1 - R2) is directly propertional to initial resistance R1 and rise in temperature (𝜃${\displaystyle 𝜃}$2 - 𝜃${\displaystyle 𝜃}$1) . So (R1 -R2)∝R1 (𝜃2-𝜃1)or, (R1 -R2)=𝛼R1 (𝜃2-𝜃1)or, 𝛼= R1 -R2

R1 (𝜃2-𝜃1)or, 𝛼= R1 -R2

R1 Δ𝜃 $$\begin{array}{c}(R_1-R_2)\propto R_1({𝜃}_2-{𝜃}_1)\\ or,(R_1-R_2)=𝛼R_1({𝜃}_2-{𝜃}_1)\\ or,𝛼=\frac{R_1-R_2}{R_1({𝜃}_2-{𝜃}_1)}\\ or,𝛼=\frac{R_1-R_2}{R_1\Delta 𝜃}\\ \end{array}$$Here, 𝛼${\displaystyle 𝛼}$ is proportionality constant known as ciefficient of resistance which is defined as increase in resistance per unit increase in temperature per degree celcius rise in temperature.Note:1. Metals have positive temperature coefficient; means resitance increases in increase in temperature.2. Semiconductors have negative temperature coefficient; means resistance decreases in increase in temperature.3. Materials like manganin, constantan etc, have almost zero temperature coefficient; means resistance does not increasse in change in temperature. Hence are used in resistance wire.ResistivityIt is observed that the resistance of a conductor is directly proportional to the lenght and inversly proportional to the corss section area of the conductor. i.e R ∝ land, R ∝ 1

Acombining both, R ∝l

Aor, R = 𝜌l

A $$\begin{array}{c}R\propto l\\ and,R\propto \frac{1}{A}\\ combiningboth,R\propto \frac{l}{A}\\ or,R=𝜌\frac{l}{A}\\ \end{array}$$Where, 𝜌 ${\displaystyle 𝜌}$is the proportionality constant, called resistivity or speccific resistance of the material of conductor. Different materials have different value of resistivity.If length(l)= 1m, area(A) = 1 m2${\displaystyle {}^2}$ then, R = 𝜌${\displaystyle 𝜌}$.Hence, the resistivity of the material of a conductor is defined as the resistance per unit length per unit area of cross – sectional of the conductor. The SI unit of electrical resistivity is the ohm⋅metre (Ω⋅m). It is commonly represented by the Greek letter ρ, rho. Conductance The reciporcal of the resistance of a conductor is known as conductance. If a conductor has resistance R, then C = 1

R$$C=\frac{1}{R}$$

Its unit is ohm

-1${\displaystyle {}^{-1}}$ or mho. ConductivityThe reciporcal of the resistivity of a conductor is called its conductivity. It is denoted by symbol σ${\displaystyle \sigma }$. If 𝜌 ${\displaystyle 𝜌}$is the resistivity of a conductor, then 𝜎= 1

𝜌 $$\begin{array}{c}𝜎=\frac{1}{𝜌}\\ \end{array}$$The unit of conductivity is Ω-1m-1${\displaystyle {\Omega }_{}^{-1}{m}^{-1}}$.Current Voltage RelationThe conductors which follow Ohms's law i.e in which I ∝ V${\displaystyle I\propto V}$, are called ohmic conductors. And those which do not follow the law is called non ohmic conductors. Most of the metals like , iron, silver , copper etc are ohmic conductors whereas electrolytes, semiconductors, diodes, tranistors etc are non ohmic semi conductors.

The I-V graph of ohmic conductor is a straight line and that of non ohmic graph is a curve.