Ohm’s Law

Ohm's Law is a key principle in electrical engineering and physics that defines the relationship between voltage, current, and resistance within an electrical circuit. Developed by the German physicist Georg Simon Ohm in the 1820s, this law offers a straightforward mathematical framework for understanding the interactions among electrical components. Serving as the cornerstone of electrical theory, Ohm's Law enables us to anticipate circuit behavior under various conditions and plays a crucial role in the design and advancement of diverse electrical devices and technologies.

1.0Statements of Ohm’s Law

In Physical quantities like temperature, pressure, volume, length, cross-section or nature of the material kept constant then current through a conductor is directly proportional to potential difference applied across it. This is called Ohm’s Law.

V ∝ I

VI=Constant=R (Resistance)

V ∝ I

V =IR

R is called the resistance of the conductor.

S.I Unit of R - Ohm

Slope =$\frac{V}{I}=\mathrm{Tan}\theta =R$

$I\propto V$

I=G V

G $\text{ is called the conductance of conductor }$

$\text{ S.I Unit of }G\Rightarrow mho\left({\Omega }^{-1}\right)\text{ or Siemens }$

$Slope=\frac{I}{V}=\mathrm{Tan}\theta =G$

2.0Proof of Ohm’s Law

From microscopic forms of Ohm’s Law

$J=\sigma E$

$\frac{I}{A}=\sigma \left(\frac{V}{l}\right)$

$\frac{V}{I}=\left(\frac{1}{\sigma }\right)\times \frac{l}{A}=\rho \times \frac{l}{A}$

$\left[\text{ Where }\rho =\frac{1}{\sigma }=\frac{m}{n{e}^2\tau }\right]$

$\frac{V}{I}=R\Rightarrow \text{ Ohm’s Law }$

$R=\frac{\rho l}{A}=\text{ Resistance }\text{ [SI Unit of }\rho \text{ is ohm }-m]$

Where $l\text{ is the length of the conductor and }$

$A\text{ is the area of cross - section, }$

$\rho \text{ is the resistivity or specific resistance. }$

3.0Microscopic Form Of Ohm’s Law

• Relation between $J,\sigma \text{ and }E$
• For an Electron q=-e

$v_d=-\frac{e\vec{E}\tau }{m}$

$\vec{J}=nq{\vec{v}}_d=n(-e)\left(-\frac{e\vec{E}\tau }{m}\right)=\frac{n{e}^2\tau }{m}\vec{E}$

$\left(\frac{n{e}^2\tau }{m}=\frac{1}{\rho }=\sigma =\text{ Conductivity of conductor }\right)$

$\vec{J}=\sigma \vec{E}\text{ or }\vec{E}=\rho \vec{J}$

4.0Experimental Verification of Ohm’s Law

• Ohm's Law can be verified through a simple experiment that demonstrates the relationship between voltage, current, and resistance in a circuit. The procedure involves measuring current through a resistor at different voltage levels using the following equipment:
• Power Supply-Variable DC supply or battery, Resistor- Known value (e.g., 100 Ω),Ammeter- Measures current (A),Voltmeter- Measures voltage (V),Connecting Wires, Breadboard  
• To set up the circuit, connect the resistor in series with the ammeter and power supply, ensuring that the voltmeter is connected across the resistor. Begin the experiment by setting the power supply to a low voltage  and recording the current (I) and voltage (V) readings. Next, gradually increase the voltage in increments  and note the corresponding current for each level. Finally, compile the data into a table to organize the recorded voltage (V) and current (I) values for analysis.
• Calculate Resistance: $\frac{V}{I}=R\Rightarrow \text{ Ohm’s Law }$, to calculate resistance for each voltage-current pair. The resistance should remain consistent.
• Graphing: Plot voltage (V) on the y-axis and current (I) on the x-axis. A straight line indicates a direct proportionality, confirming Ohm's Law.
• The experiment should demonstrate that as voltage increases, current also rises linearly, validating Ohm's Law and emphasizing the predictable nature of electrical circuits.

5.0V – I Characteristics (Linear and Non-linear)

1. Linear V-I characteristics

• At fixed temperature, current is directly proportional to the applied potential difference. This law is called ohm's law and substances which obey it are called ohmic or linear conductors

2. Non- Linear V-I characteristics

• The relation between V and I relies on the signifier of V. In other words, if I is the current for a certain V, then flipping the direction of V keeping its strength fixed, does not produce a current of the same magnitude as I in the opposite direction (Figure 1).

• The relation between V and I is not distinctive, i.e., there is more than one value of V for the same current I (Figure 2). A material exhibiting such a behaviour is GaAs.

6.0Limitations of Ohm’s Law

• The straight line V-I graph does not pass through the origin.
• V-I relationship is non-linear
• V-I relationship leans on the sign of V for the same absolute value of V
• V-I relationships are not unique.
• Examples of non-ohmic conductors are water voltameter, p-n junction diode, thyristor

7.0Applications of Ohm’s Law

• It aids students in calculating voltage, current, and resistance in simple series and parallel circuits, allowing them to understand how components function under various conditions.
• It is used to design basic electrical circuits, calculating the necessary resistor values to ensure household appliances operate safely and efficiently.
• Ohm's Law is essential for electrical safety, as it helps to calculate the maximum allowable current in circuits to avoid overheating and hazards.

8.0Ohm’s Law Triangle

• The Ohm's Law triangle is a useful visual aid that illustrates the relationship between voltage (V), current (I), and resistance (R) in electrical circuits. Arranged in a triangular format, voltage is placed at the top, while current and resistance are positioned at the bottom corners. 
• Voltage (V): Multiply current (I) by resistance (R):V=I R
• Current (I): Divide voltage (V) by resistance (R):$I=\frac{V}{R}$
• Resistance (R): Divide voltage (V) by current (I): $R=\frac{V}{I}$

9.0Sample Questions on Ohm’s Law

Q-1.In the shown I-V Curve Calculate $\frac{R_1}{R_2}$.

Solution: Slope of given graphs gives conductance (G)

$\mathrm{Tan}60^{\circ }=G_1=\sqrt{3}=\frac{1}{R_1},R_1=\frac{1}{\sqrt{3}}$

$\mathrm{Tan}30^{\circ }=G_2=\frac{1}{\sqrt{3}}=\frac{1}{R_2},R_2=\sqrt{3}$

$\frac{R_1}{R_2}=\frac{\frac{1}{\sqrt{3}}}{\sqrt{3}}=\frac{1}{3}$

Q-2.The voltage-current graphs for two resistors of the same material and the same radius with lengths,L₁ and L₂ are shown in figure. If L₁ > L₂, state with reason which of these graphs represents voltage current change for L₁.

Solution: As, $R_A=\frac{V}{I_A}$   and  $R_B=\frac{V}{I_B}$

$\frac{I_A}{I_B}=\frac{R_B}{R_A}$

Fom figure as $I_A>I_{B^{\prime }}{R}_B>R_A$ Since $R\propto L$ Graph B represents the voltage current change for $L_1$

Q-3.The I-V characteristics of a resistor is monitored to deviate from a straight line for higher value of current as shown in figure Why?

Solution: For higher value of current, the resistor gets heated and consequently its resistance increases. The resistor becomes non-ohmic due to which the I-V characteristic deviates from the straight line thereby showing lesser current for the same voltage.