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Electric Current and Ohm's Law

Electric current is the rate at which charge flows through a surface:

Electric current illustration

If a charge QQ flows through in a time interval tt, the average current is:

Iavg=Qt.I_{avg} = \frac{Q}{t}.

In the limit t→0t \to 0, the instantaneous current is equal to:

I=Qt.I = \frac{Q}{t}.

The unit of electric current is ampere (AA).

Suppose a cylindrical conductor with cross sectional area AA with nn charges per unit volume. The charges have equal magnitude qq flowing at a velocity vd\boldsymbol{v_d} (known as the drift velocity - velocity at which charges move when an external electric field is applied):

Electric current density illustration

the current is equal to:

I=∬SJ⋅dA,I = \iint_S \boldsymbol{J} \cdot d\boldsymbol{A},

where A\boldsymbol{A} is the vector pointing perpendicular to the cross sectional area.

Suppose that in time tt the charges move by x=vdtx = v_d t, then Q=qnAx=qnAvdtQ = q n A x = q n A v_d t and the average current is equal to:

Iavg=Qt=qnAvd. \begin{align*} I_{avg} &= \frac{Q}{t} \\ &= q n A v_d. \end{align*}

Current density is the current per unit area:

J=nqvd,\boldsymbol{J} = nq \boldsymbol{v_d},

J\boldsymbol{J} and vd\boldsymbol{v_d} point in the same direction for positive charges and in opposite direction for negative charges.

Next, we will find the drift velocity vd\boldsymbol{v_d}. By Newton's second law, an electron experiences an acceleration equal to:

a=Feme=−eEme, \begin{align*} \boldsymbol{a} &= \frac{\boldsymbol{F_e}}{m_e} \\ &= -\frac{e \boldsymbol{E}}{m_e}, \end{align*}

where e≈1.6⋅10−19 Ce \approx 1.6 \cdot 10^{-19}\ C is the elementary charge.

Let the velocity of an electron immediately after a collision be vi\boldsymbol{v_i} the velocity of the same electron immediately before the next collision is equal to:

vf=vi+at=vi−eEmet. \begin{align*} \boldsymbol{v_f} &= \boldsymbol{v_i} + \boldsymbol{a}t \\ &= \boldsymbol{v_i} - \frac{e \boldsymbol{E}}{m_e}t. \end{align*}

The average of vf\boldsymbol{v_f} is equal to:

⟨vf⟩=⟨vi⟩−eEme⟨t⟩,\langle\boldsymbol{v_f}\rangle = \langle\boldsymbol{v_i}\rangle - \frac{e \boldsymbol{E}}{m_e} \langle t\rangle,

which is equal to the drift velocity.

In the absence of electric field the motion is completely random, where ⟨vi⟩=0\langle\boldsymbol{v_i}\rangle = \boldsymbol{0}. If ⟨t⟩=τ\langle t\rangle = \tau, the average characteristics time between collisions, the drift velocity is equal to:

vd=−eEmeτ. \begin{align*} \boldsymbol{v_d} &= - \frac{e \boldsymbol{E}}{m_e} \tau. \end{align*}

The current density is then equal to:

J=−nevd=ne2τmeE.\boldsymbol{J} = -n e \boldsymbol{v_d} = \frac{n e^2 \tau}{m_e} \boldsymbol{E}.

Note: J\boldsymbol{J} and E\boldsymbol{E} will always be in the same direction.

The current density is usually expressed as:

J=σE,\boldsymbol{J} = \sigma \boldsymbol{E},

where σ=ne2τme\sigma = \frac{n e^2 \tau}{m_e} is called the conductivity of the material. The above equation is known as the (microscopic) Ohm's law.

Suppose a cylindrical conductor of length ll with uniform electric field E\boldsymbol{E} and with cross sectional area AA:

Electric current illustration

The voltage is equal to:

U=Vb−Va=−∫abE⋅ds=El, \begin{align*} U &= V_b - V_a \\ &= -\int_a^b \boldsymbol{E} \cdot d\boldsymbol{s} \\ &= El, \end{align*}

implying E=UlE = \frac{U}{l}. The magnitude of the current density is equal to:

J=σUl,J = \sigma \frac{U}{l},

implying U=JlσU = \frac{J l}{\sigma}, where J=IAJ = \frac{I}{A}:

U=IlAσ=RI,U = \frac{I l}{A \sigma} = RI,

where R=lσAR = \frac{l}{\sigma A} is the resistance of the conductor. The equation U=IRU = IR is the "macroscopic" version of the Ohm's law. The unit of resistance is ohm (Ω\Omega).

The resistivity ρ\rho of a material is the reciprocal of conductivity:

ρ=1σ=mene2τ.\rho = \frac{1}{\sigma} = \frac{m_e}{n e^2 \tau}.

Using E=UlE = \frac{U}{l} and J=IAJ = \frac{I}{A}, the resistance can be derived:

J=Eρ  ⟹  ρ=EJ,ρ=UAIl=RAl,R=ρlA. \begin{align*} J &= \frac{E}{\rho} \implies \rho = \frac{E}{J}, \\ \rho &= \frac{UA}{Il} \\ &= \frac{RA}{l}, \\ R &= \frac{\rho l}{A}. \end{align*}

The resistivity varies with temperature TT:

ρ=ρ0[1+α(T−T0)],\rho = \rho_0 [1 + \alpha(T - T_0)],

where α\alpha is the temperature coefficient of resistivity, ρ0\rho_0 is the resistivity at the temperature T0T_0. Below is a table of some materials at 20 ∘C20\ ^{\circ} C:

Materialρ0\boldsymbol{\rho_0}σ0\boldsymbol{\sigma_0}α\boldsymbol{\alpha}
Silver1.59⋅10−81.59 \cdot 10^{-8}6.28⋅1076.28 \cdot 10^{7}3.8⋅10−33.8 \cdot 10^{-3}
Copper1.71⋅10−81.71 \cdot 10^{-8}5.81⋅1075.81 \cdot 10^{7}3.9⋅10−33.9 \cdot 10^{-3}
Aluminum2.82⋅10−82.82 \cdot 10^{-8}3.54⋅1073.54 \cdot 10^{7}3.9⋅10−33.9 \cdot 10^{-3}
Iron1⋅10−71 \cdot 10^{-7}1⋅1071 \cdot 10^{7}5⋅10−35 \cdot 10^{-3}
Platinum1.06⋅10−71.06 \cdot 10^{-7}9.43⋅1069.43 \cdot 10^{6}3.9⋅10−33.9 \cdot 10^{-3}
Carbon3.5⋅10−53.5 \cdot 10^{-5}2.85⋅1042.85 \cdot 10^{4}−5.01⋅10−4-5.01 \cdot 10^{-4}
Silicon6.4⋅1026.4 \cdot 10^{2}1.56⋅10−31.56 \cdot 10^{-3}−7.5⋅10−2-7.5 \cdot 10^{-2}
ρ=1.68⋅10−8 ΩmA=3.14⋅10−6 m2R=1.61⋅10−2 Ω \begin{align*} \rho &= 1.68 \cdot 10^{-8}\ \Omega m \\ A &= 3.14 \cdot 10^{-6}\ m^2 \\ R &= 1.61 \cdot 10^{-2}\ \Omega \end{align*}

Consider a circuit with a battery and a resistor of resistance RR:

Electric current illustration

The voltage between the points aa and bb is U=φb−φa>0U = \varphi_b - \varphi_a > 0. If a charge qq is moved from aa to bb through the battery, its electric potential energy is increased by:

ΔEp=qU.\Delta E_p = q U.

The charge qq is unchanged upon returning to aa. When the charge moves through the resistor, its potential energy is decreased. The rate of energy loss is equal to:

P=ΔEpt=qtU=IU, \begin{align*} P &= \frac{\Delta E_p}{t} \\ &= \frac{q}{t}U \\ &= IU, \end{align*}

this is the power supplied by the battery. Using U=IRU = IR, the equation can be rewritten:

P=I2R=U2R.P = I^2 R = \frac{U^2}{R}.