Dimensions of Resistance: Formula, Derivation, and Examples

The Dimensions of Resistance are a standard topic in the JEE Main Physics syllabus, crucial for understanding how electrical properties relate to fundamental quantities like mass, length, time, and electric current. Mastering this dimensional formula helps students verify equations and derive SI units, two common exam requirements. Vedantu regularly emphasises this connection to dimensional analysis and equation checking.

In Physics, resistance measures how strongly a material opposes electric current. Its correct SI unit is the ohm (Ω). To solve problems and avoid mistakes in the exam, students must know both the units and the dimensional formula of resistance.

Dimensions of Resistance and the Ohm’s Law Connection

The Dimensions of Resistance come directly from Ohm’s law, which states V = I × R. Here, V is voltage in volts, I is current in amperes, and R is resistance in ohms. Understanding these relationships allows you to derive other related concepts such as the dimension of inductance and dimensions of capacitance.

The voltage itself is defined in terms of work done per unit charge, making it necessary to recall the dimensional formula for work and charge for a rigorous derivation.

Step-by-Step Derivation: Dimensions of Resistance

1. Recall Ohm’s law: V = I × R.
2. Rearrange: R = V / I.
3. Voltage (V) = Work (W) / Charge (Q).
4. Work W = Force (F) × distance, and Force = m × a = M L T^-2.
5. So, dimensions of Work: M L² T^-2.
6. Charge Q = Current × Time = I T.
7. Therefore, dimensions of Charge: I T.
8. Voltage V (W/Q): (M L² T^-2) / (I T) = M L² T^-3 I^-1.
9. Substitute into R = V/I: M L² T^-3 I^-1 ÷ I = M L² T^-3 I^-2.
10. Final dimensional formula of resistance: M¹ L² T^-3 I^-2.

Applications and Importance of Dimensions of Resistance

The dimensions of resistance help students quickly check if expressions involving resistance are correct in numerical problems. This process is called dimensional analysis. It also helps verify formulas when solving for resistance in combinations or circuits.

• Check if circuit equations are dimensionally valid.
• Avoid calculation errors by confirming units of unknowns.
• Compare resistance with related concepts like resistivity.
• Screen for exam traps demanding an exact dimensional formula.
• Clarify confusion between dimensions of resistance and other electrical quantities.

Common Mistakes and Fast Revision for Dimensions of Resistance

• Forgetting current’s exponent is -2 not -1.
• Writing L in place of L²; always note R = M L² T^-3 I^-2.
• Confusing units (ohm Ω) with dimensions (M¹ L² T^-3 I^-2).
• Using charge or voltage incorrectly in derivations.
• Mistaking dimensional formula for capacitance or inductance in exam shortcuts.

Here is a concise worked example for rapid practice. This matches JEE question style and avoids common traps.

1. Given R = V / I; Voltage V (M L² T^-3 I^-1), Current I (I).
2. Divide: (M L² T^-3 I^-1) ÷ (I) = M¹ L² T^-3 I^-2.
3. Therefore, the dimensions of resistance are M¹ L² T^-3 I^-2.

Understanding dimensions of resistance also lets you check the validity of equations like those for series and parallel circuits, Ohm’s law, or Kirchhoff’s laws. It forms a core principle for error spotting in measurement errors and quick-checks during time-pressured tests.

• Review the difference between resistance and resistivity for exam clarity.
• Always double-check exponents in the final dimensional formula.
• Revise using summary pages for units and dimensions.
• Practice circuit numericals for a range of resistance values.

With regular practice and the correct dimensional formula, students maximise scores in questions on dimensions of resistance, reinforce core concepts for current electricity, and avoid mixing up related terms like impedance or capacitance. For full mastery, use expert-reviewed study material from Vedantu on all dimensions and units.