What is Ohm’s law?

Ohm’s law is a prominent concept in physics and electronics. It gives the relation between the current and the voltage. It is used to analyze and construct electrical circuits. Ohm's law states that the voltage across a conductor is directly proportional to the current flowing through it.

Introduction to Ohm's law

Ohm’s law was published by physicist Georg Ohm in 1827. Ohm used a thermocouple device as a voltage source and he used a galvanometer to measure the quantity of current. The thermocouple device generates a voltage that depends upon the temperature. Ohm observed that the galvanometer reading is proportional to temperature. He used conducting wires of various materials; the obtained result was similar.

Understanding current and voltage

The electric current is nothing but the transmission of charges through the electric conductor. We can classify objects as electric conductors and insulators. In an electric conductor, the current can flow easily whereas, in insulators, electric current cannot be transmitted.

There are two types of charges in an atom, they are protons which are positively charged and negatively charged electrons. The electric current is described as the conventional flow of electric charges. Hence, the current is defined as the rate of flow of electric charges per unit time. The quantity current gives the number of charges transferred per second (or unit time) and is denoted by the letter I. Amperes (A) is the unit of current.

The transmission of positively charged protons does not take place as the proton is present in the nucleus and held by the nuclear force.

Consider, a simple example where an electric battery is made use to glow a bulb. As the battery has connected to a bulb, the bulb glows as the current passes through it. The energy to push the electrons through the bulb was stored in the battery. The force required to push the sea of electrons is called electromotive force (emf); which is also termed as voltage. Voltage is the work done in bringing a unit charge from one point to another and is expressed by V (in volts).

Ohm’s law statement

Ohm’s law states that the voltage across a conductor is directly proportional to the current flowing through it, if physical quantities like length, area of cross-section, and temperature remain constant.

Ohm’s law is expressed as follows,

$V\propto I$

$V=IR$  (i)

Where R is a constant.

R is the quantity, denotes the resistance, which is the measure of opposition to the transmission of electric current. The unit is ohms (Ω).

Equation (i) is the mathematical form of ohm's law.

A conductor with a high value of resistance opposes electric current effectively but materials with low resistance do not oppose electric current effectively.

The insulators which hardly conduct electricity have very high quantity of resistance.

In the above circuit, voltage V is present across the resistor R. A resistor is a device that opposes the transfer of electric current. To measure the voltage across the resistor, a voltmeter was installed across the resistor. As supplied current I is increased through the resistor, the voltage across the resistor also increases.

Resistance of any material depends upon various factors like the nature of the material and temperature. It is also affected by the dimension-like area of cross-section and length. For example, consider any conducting wire of a certain length. If R is the resistance of the wire. If the wire is split into equal halves. The resistance offered by the conducting wire becomes R/2 i.e. the quantity resistance is proportional to the length of the given wire.

Let l be the length of the wire. The resistance is directly proportional to the length of the wire. Hence, we can write,

$R\propto I$   (ii)

The resistance offered by the conducting wire also depends on the area of cross-section. Consider two wires with the area of cross-section A and 2A, if both the wires are of the same material and if the resistance of first wire is R, then the resistance of the second wire will be R/2 .

∴ $R\propto \frac{1}{A}$   (iii)

As area increase the resistance offered by the conducting wire decrease.

From equation (i) and equation (iii),

$R\propto \frac{l}{A}$

or

$R=\rho \frac{l}{A}$

The equation for resistivity becomes,

$\rho =R\frac{A}{l}$

ρ is called resistivity (in Ωm), which is a constant for a specific material. Resistivity is not related to the size and shape of the conductor. Hence, two wires of different lengths or different areas have different resistance but, the resistivity is the same for any wire of the same material. The resistivity depends upon the temperature, as temperature increases the resistivity of the electric conductors also increases.

The quantity ρ is the conductivity which is denoted by (unit is ${\Omega }^{-1}{m}^{-1}$ ).

Hence, materials with the least resistance will have maximum conductivity.

Ohm’s law in electrical circuits

Let us apply ohm’s law to the circuits of different configurations.

Resistors in series

Let V_B be the voltage of dc source that is connected to resistors with resistance R₁ and R₂ in series i.e. the end of the resistor R₁ attached to the opposite end of R₂ remaining ends are connected to electrodes of the battery. If I is the electric current through the resistor, from Kirchhoff’s law, one can conclude that the current through both the resistors remains the same.

From ohm’s law, the voltage across R₁ is,

$V_1=I{R}_1$

The voltage across R₂ is,

$V_2=I{R}_2$

Then the net voltage is,

$V_{net}=I{R}_1+I{R}_2$

$V_{net}=I(R_1+R_2)$

If the resistors R₁ and R₂ are replaced by resistance R_s,

$V_{net}=I{R}_s$

$R_s=R_1+R_2$

The equivalent resistance is the sum of the resistance of the individual series resistors.

Say the two resistance in series are 5Ω and 4Ω. The circuit behaves similar to a circuit with the same supplied voltage and single 9Ω resistor ∵$R_s=5\Omega +4\Omega =9\Omega$.

Resistors in parallel

If the resistors are arranged parallelly as shown below.

The system of resistors is arranged in parallel If both resistors are attached and the common ends are attached to the battery. If the resistances of resistors are R₁ and R₂, so the current transmitting through each component is I₁ and I₂.

The voltage V_B remains the same across the resistors.

From ohm’s law,

$V=IR$

The current is,

$I=I_1+I_2$

$I=\frac{V}{R_1}+\frac{V}{R_2}$

Put $\frac{1}{R_P}=\frac{1}{R_1}+\frac{1}{R_2}$

$I=\frac{V}{R_P}$

Hence the net resistance of the system of resistors arranged parallel is $\frac{1}{R_1}+\frac{1}{R_2}$.

If resistances of 5Ω and 4Ω are arranged in parallel then the equivalent resistance is,

$\frac{1}{R_P}=(\frac{1}{5}+\frac{1}{4}){\Omega }^{-1}=\frac{9}{20}{\Omega }^{-1}=0.45{\Omega }^{-1}$

$R_P=2.22 \Omega$

The system of resistors 4 Ω and 5 Ω in parallel exhibit equivalent resistance of 2.22 Ω.

Formulas

The formula that relates voltage and current is,

$V=IR$

Where V is voltage (in volts).

I is current (in ampere).

R denotes resistance (in Ω).

The resistivity is,

$\rho =R\frac{A}{l}$

Where, A is the area of cross-section.

l is the length of the conductor.

Net resistance, if the resistors are connected in series is,

$R_s=R_1+R_2$

Net resistance, if the resistors are connected in parallel is,

$R_P=\frac{R_1{R}_2}{R_1+R_2}$

Context and Applications

This topic is significant in physics for both undergraduate and graduate courses, especially for bachelors and masters in science (physics), and bachelors in technology (electrical engineering).

Practice Problems

Question 1: An circuit was constructed by connecting a battery labeled 12V to resistors 6Ω, 3Ω, and 2Ω  in series. What is the amount of current passing through the resistors?

(a) 0.43 A

(b) 4.54 A

(c) 1.09 A

(d) 2.45 A

Answer: The correct option is c.

Given data:

$R_1=6\Omega$

$R_2=3\Omega$

$R_3=2\Omega$

Explanation:

The equivalent resistance is,

$R=R_1+R_2+R_3$

$R=6\Omega +3\Omega +2\Omega =11\Omega$

From ohm's law,

$V=IR$

or

$I=\frac{V}{R}$

$I=\frac{12}{11}A=1.09A$

The electrical current through the resistor is 1.09 ampere.

Question 2: Resistivity of a conductor depends upon ____ of the conductor.

(a) Area of cross-section

(b) Length

(c) Nature of material

(d) The voltage at the ends

Answer: The correct option is c.

Explanation: The resistivity of a material is independent of its length and area. It only depends upon the nature of the material. Since is constant even when A and l change.

Question 3: In a circuit, if the resistors are connected in parallel then the voltage across each resistor _____.

(a) Remains the same

(b) Completely different

(c) Depends upon the resistors

(d) Depends upon current

Answer: The correct option is a.

Explanation: From Kirchoff's law, the voltage across the resistors in parallel remains the same.

Question 4: The direction of flow of current is _____.

(a) Same as the flow of electrons                           (b) Opposite to the flow of electrons

(c) Depends upon the magnitude voltage       (d) Depends upon the magnitude of current

Answer: The correct option is b.

Explanation: The direction of current is the conventional direction of the flow of electrons.

Question 5: Usually the conductivity of a metal is ___ a non-metal.

(a) More than            (b) Less than

(c) Same as                (d) None of the above

Answer: The correct option is a.

Explanation: The metals have got very low resistivity than the non-metals hence the conductivity is more than the conductivity of non-metal.