Ohm’s Law and Resistance - Definition, Formulas, Applications & Problems

What Is Ohm’s Law?

At the heart of it, Ohm’s law states that the electric current flowing through a conductor between two points is directly proportional to the voltage across these two points.

The formula for Ohm’s law is represented as;

V ∝ I

Therefore, we can rewrite this as V = RI, where R is a constant known as resistance. The value of R depends on the dimensions and the material of the conductor. Its SI unit is Ohm (Ω).

 

Ohm’s law was established through numerous experiments, somewhat akin to the laws of thermodynamics . It's of immense significance and finds its application across all branches of electronic studies or science. The law is instrumental in carrying out various calculations, such as determining the value of resistors, the current in a circuit, or measuring voltage.

Furthermore, Ohm’s law assists us in understanding how current flows through materials like electrical wires, and so on.

Also Read: Electric Current

Ohm’s Law Formulas

Essentially, there are three types of Ohm’s law formulas or equations. They are;

• I = V / R
• V = IR
• R = V / I

Here, I represents the current, V stands for voltage, and R is the resistance.

Ohm’s Law Triangle

Ohm’s law is a staple in the study of electronics and electricity. Hence, it's crucial for students to remember the formulas as they aid in circuit analysis. The formulas of Ohm’s law can be easily memorized using the Ohm’s law triangle. This triangle helps us to effortlessly represent the interchangeability of the equations. By dividing a triangle into three parts, we can assign the values, V, I, R into the triangle. V on top, I on the left side, and R on the right side. It will appear like this;

 

Whenever you want to solve a calculation, you can simply cover the value you are looking for.

Vector Form of Ohm’s Law

The vector form of Ohm’s law is used in the fields of electromagnetics and material science. The vector form is given as,

Where,

^– represents the vector

The conductivity is the reciprocal of resistivity. This leads us to the next question, what exactly is resistivity?

Resistivity

Let's consider a conductor with a cross-sectional area ‘a’ and length ‘l’. Let its resistance be R. Let it have a voltage difference of V volts between its end and a current I flowing through it.

Now consider another conductor of the same dimensions, except its length is doubled. This can be considered as two conductors of the first kind connected in series. Therefore, their voltages add up. The voltage here will, therefore, become 2V.

From Ohm’s law, R = (V/I)

Here, R = (2V / I) = 2R

This implies that resistance increases with length. That is, R ∝ l

Concept point: This does make sense if you think about it. If the length increases, then the flowing electrons will face more obstacles, and hence resistance increases.

Similarly, now consider a conductor with half the cross-sectional area as the first conductor. This can be considered as one of the two conductors in a parallel connection. Therefore, the current becomes half of the original value I/2. Thus, the value of the current becomes I/2 now.

Here, R = (V/ (I/2)) = 2R

This means that resistance increases when the area of the cross-section decreases. That is,

R ∝ 1/A

Also Read: Dimensional Formula of Resistance

Concept point : As the area of the cross-section decreases, the flowing electrons come into more contact with positive nuclei of metals. Thus, resistance increases.

Combining these two results we get,

R ∝ l/A

Or, R = ρl / A where ρ is a constant called the coefficient of resistance or resistivity.

In the above equation if L = 1m and A = 1m ² , then R = ρ.

This signifies that resistivity is the resistance of a conductor with 1m length and 1m ² area. The resistivity of a conductor is dependent on the nature of its material and certain external factors, such as heat.

Vector Expression

V = IR

⇒ V = I ρL/A

⇒ V = L ρI/A

⇒ V = JρL since current density J = I/A

⇒ El = JρL since V = El

Cancelling L from both sides, E = Jρ or J = E/ρ

1/ρ = σ which is conductivity.

Therefore, = σ where the bar letters are vectors.

Concept point: You might be wondering why current is a scalar unit while current density is a vector unit. To understand this, you need to remember how a vector is defined. A vector is a quantity with both magnitude and direction that obeys the laws of vector addition. The current does not follow the rules of vector addition and hence, it is not a vector.

The broader aspect of this concept that you need to understand here is that vector quantities are concepts we use to aid our calculations. However, sometimes we have to make exceptions in order to make our calculations easier or even correct in some cases.

Limitations of Ohm’s Law

Despite its widespread application, Ohm’s law does have some limitations. They are as follows:

• Ohm’s law is an empirical law which is found to be true for most experiments but not for all.
• Some materials do not obey Ohm’s law under a weak electric field.
• Ohm’s law holds true only for a conductor at a constant temperature. Resistivity changes with temperature.

Joule’s heat is given by H = I ² Rt where I is current, R is resistance and t is time. As long as the current flows, the temperature of the conductor will increase.

• Ohm’s law is not applicable to in-network circuits.
• Ohm’s does not apply directly to capacitor circuits and inductor circuits.
• The V-I characteristics of diodes are different.

 

• The V-I graph of ohmic conductors is not a straight line. It shows some variation.

Also Read: Types of Circuits

Applications of Ohm’s Law

Ohm’s law is incredibly useful and has a plethora of applications. A few are listed below:

• It is widely used in circuit analysis.
• It is used in devices such as ammeters, multimeters, etc.
• It is used in the design of resistors.
• It is used to get the desired voltage drop in circuit design.
• Advanced laws such as Kirchhoff’s Norton’s law, Thevenin’s law are based on Ohm’s law.
• Electric heaters, kettles and other types of equipment operate based on Ohm’s law.
• A laptop and mobile charger using DC power supply in operation and the working principle of DC power supply depend on Ohm’s law.

Ohm’s law also has many other uses as well.

Analogies of Ohm’s Law

There are some interesting analogies to Ohm’s law:

Hydraulic Analogy

A hydraulic circuit can be compared to an ohmic conductor to simplify the problem. Here, pressure functions similarly to voltage and flow rate operates similarly to current.

Temperature Analogy

Similarly, a temperature circuit can also be compared to an ohmic conductor. Here, the temperature gradient operates similarly to voltage and heat flow operates similarly to current.

In fact, GS Ohm discovered Ohm’s law by drawing an analogy from Fourier’s work on heat flow.

Definitions of Current, Voltage and Resistance

Apart from the standard definitions, it's essential to know what these terms actually mean. Let’s start with the current. More specifically, the electric current. What comes to your mind when you first hear the word current? Perhaps a river that is flowing. That’s what electric current is, it can be visualised as the flow of electrons from one place to another.

Now, why does the flow occur? Well, a river flows from the mountains to the sea, i.e., from a high place to a low place. Similarly, current flows from a high potential to a low potential, and this change of potential is called voltage.

You may notice that every river has some obstructions and that’s what resistance is to current. Resistance is the property of the conductor that obstructs the flow of current.

Ohm’s Law Problems and Solutions

1. We have an unknown resistor which dissipates 30 watts power while dropping 15 volts across it. Find the current that is flowing through it.

Solution:

We apply Ohm’s law formula for I,

I = P/V = 30/15 = 2 A

 

2.  Apply Ohm’s law for the following circuits.

 

Solution:

Applying Ohm’s law on the circuit (a)

I = 3 A

Applying Ohm’s law on the circuit (b)

V = I × R

V = 1 × 13

V = 13 V

Applying Ohm’s law on the circuit (c)

R = 2 Ω

3. How much voltage will be dropped across a 50 kΩ resistance whose current is 300 µA?

Given

R = 50 kΩ = 50 × 10 ³ Ω

I = 300 µA = 300 × 10 ^-6 A

Solution:

V = I × R

V = (300 × 10 ^-6 ) × (50 × 10 ³ )

V = 15 V

4. Apply the power (P) formula for the following circuit.

 

Given:

I = 6 A

V = 30 V

R = 5 Ω

Solution:

Power formula: P = VI or P = I ² R or

P = 30 × 6

P = 180 W

5. An electric circuit has a current of 3.00 A flowing through it. Find the potential difference across a resistor, with a resistance of 250 Ω.

Solution:

Using the Ohm’s Law formula:

V = IR = (3.00 A)(250 Ω)

V = 750V

The potential difference across the resistor in the circuit is 750 V.