Ohm’s Law: Definition, Formula, Limitations, Derivation, Diagram and Deduction of Ohm's Law

Ohm's Law

According to Onm's Law if there is no change in the physical state of the conductor (length, temperature, nature of matter, and cross-sectional area) then the current flowing through a conductor is directly proportional to the applied potential difference across the ends of a conductor.

`I \prop V`

`V \prop I`

`V = R I`

Where R is a proportionality constant and is called the resistance of the conductor.

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

Where

V = Voltage across the ends of a conductor (Measured in volts),

I = Current (Measured in amperes),

R = Resistance (Measured in ohms).

This fundamental equation of Ohm's Law provides a simple but powerful way to understand the behavior of electrical circuits and is a cornerstone of electrical engineering and physics.

* S.I. unit of resistance is ohm which is denoted by `\Omega`.

* `1 \Omega = \frac {1 \text {volt}}{1 \text {ampere}}`

* Dimension of resistance R is `[M^1L^2T^{- 3}A^{- 2}]`

1 Ohm's Definition

According to Onm's Law

`I \prop V`

`V \prop I`

`V = R I`

Where R is a proportionality constant and is called the resistance of the conductor.

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

If V = 1 volt and I = 1 ampere

`R = \frac {1 \text {volt}}{1 \text {ampere}}`

`R = 1 \Omega`

If a 1-volt potential difference is applied across the ends of the conductor and 1-ampere current flows through it then the resistance of the conductor will be `1 \Omega`

I - V Curve in Ohm's Law

If we plot a graph between current I (on the y-axis) and voltage V (on the x-axis) we get a straight line.

Here, `tan\theta` is the slope of the curve.

`tan\theta = \frac{I}{V} = \frac {1}{R}`

Resistance

Resistance of a conductor is a property of a conductor to oppose the flow of charge through it. It is given by the ratio of the potential difference across the ends of a conductor to the current flowing through the conductor.

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

Where

R = Resistance

V = potential difference across the ends of a conductor

I = electric current

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

If V = 1 volt and I = 1 ampere

`R = \frac {1 \text {volt}}{1 \text {ampere}}`

`R = 1 \Omega`

Resistance of a conductor is said to be 1 ohm if the potential difference of 1 Volt across the ends of the conductor makes the current of 1 ampere to flow through it.

Electric potential difference is independent of the resistance whereas, the current is inversely proportional to the resistance.

Resistivity

The resistance (R) of the conductor is directly proportional to the length.

`R \prop l`

The resistance (R) of the conductor is inversely proportional to the cross-section area (A) of the conductor.

`R \prop \frac{1}{A}`

From eq. 1 and eq. 2

`R \prop \frac{l}{A}`

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

Where `\rho` is the proportionality constant and is known as resistivity. It depends on the material of the conductor but not on its dimensions.

According to Ohm's law

`V = I \times R`

`V = I \times \rho \frac{l}{A}`

`V = J \times \rho \times l`

`E \times l = J \times \rho \times l`

`E = J \times \rho `

`E = \rho \times J `

This equation often states Ohm's law.

Here, `\frac{1}{\rho} = \sigma` where `\sigma` is called the conductivity.

Where,

I = Current

` J = \frac{I}{A}` = current density (current per unit area)

The SI units of the current density are `\frac{A}{m^2}`

`\rho` = resistivity of the conductor

E = Magnitude of uniform electric field

`l` = Length of conductor

V = Potential difference across ends of the conductor

Deduction of Ohm's Law

We know that the relation between drift speed and electric current is

`I = n A e v_d`

`\frac{I}{A} = n e v_d`

` J = n e v_d` Where J is the current density

` J = n e \frac{e \tau}{m} E`

` J = \frac{n e^2 \tau}{m} E`

Here, m, e are constant, and n, `\tau` are the characteristics of the conductor.

For homogeneous conductor ` \frac{n e^2 \tau}{m} ` is constant. It is called the conductivity `sigma` of the matter.

Thus,

` J = \sigma E`

In vector form

` \vec J = \sigma \vec E`

This equation is the microscopic form or vector form of Ohm's law.

Consider a conductor having a length of l and a cross-sectional area A.

Now we know that

` J = \sigma E`

` \frac {I}{A} = \sigma \frac{V}{l}`

` V = \frac{1 }{ \sigma}\frac {l}{A} I`

` V = \rho \frac {l}{A} I`

` V = \frac {\rho l}{A} I`

` V = R I`

This is the Ohm's law.

Where,

`\rho = \frac {1}{\sigma}=` Resistivity

` \sigma =` Conductivity

`E = \frac {V}{l} = `electric field inside the conductor.

Limitations of Ohm's Law

Non-Linear Devices

Semiconductors, diodes, and transistors do not strictly follow Ohm's Law. These devices exhibit non-linear behavior and are called non-ohmic. The graph between voltage (V) and current (I) is not a straight line.

Heating effect due to High Current

If a large current flows through a conductor. According to `H = I^2 R t` heating takes place, due to which the resistance of the conductor also increases because of high resistance, instead of getting a straight line we get the curve.

Temperature-Dependent Resistance

Ohm's law is not applicable at high temperatures because resistance increases with temperature.

Applications of Ohm's Law in Electrical Circuits

Series Circuits

In a series combination of resistors, the current remains the same through each component, while the voltage is distributed among them.

Parallel Circuits

In a parallel combination of resistors, the voltage remains the same through each component, while the current is distributed among them.

Mixed Circuits

In a series and parallel mixed combination of resistors, Ohm's law is used to find individual branch currents, total resistance, and voltage drops.

Other Applications

1. Electrical fuse ratings.

2. Power consumption calculations.

3. Fan speed regulation.

4. Resistor selection.

5. Circuit parameter determination like voltage, resistance, and current.

Ohm's Law Solved Example

The resistance of an electric coil is 60 `\Omega` and a current of 3.2 A flows through the resistance. Find the voltage between two points. (Ans. V =192 V)