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Notes • Solved Problems

Units and Dimensions class 11

By Sanket Barde • Physics • JEE

Contents

• Mechanical Quantities
• Electrical Quantities
• Magnetic Quantities
• Thermal Quantities
• Wave and Oscillation Quantities
• Optics Quantities
• Modern Physics Quantities
• More JEE tutorials

I. Mechanical Quantities

1. Displacement

\[ x = \text{distance moved} \]

\[ [x] = L \]

2. Velocity

\[ v = \frac{x}{t} \]

\[ [v] = \frac{L}{T} = LT^{-1} \]

3. Acceleration

\[ a = \frac{v}{t} \]

\[ [a] = \frac{LT^{-1}}{T} = LT^{-2} \]

4. Force

\[ F = ma \]

\[ [F] = M(LT^{-2}) = MLT^{-2} \]

5. Momentum

\[ p = mv \]

\[ [p] = M(LT^{-1}) = MLT^{-1} \]

6. Impulse

\[ J = Ft \]

\[ [J] = (MLT^{-2})(T) = MLT^{-1} \]

7. Work / Energy

\[ W = Fs \]

\[ [W] = (MLT^{-2})(L) = ML^{2}T^{-2} \]

8. Power

\[ P = \frac{W}{t} \]

\[ [P] = \frac{ML^{2}T^{-2}}{T} = ML^{2}T^{-3} \]

9. Pressure

\[ Pressure = \frac{F}{A} \]

\[ [P] = \frac{MLT^{-2}}{L^{2}} = ML^{-1}T^{-2} \]

10. Density

\[ \rho = \frac{m}{V} \]

\[ [\rho] = \frac{M}{L^{3}} = ML^{-3} \]

11. Angular Displacement

\[ \theta = \frac{\text{arc}}{\text{radius}} \]

\[ [\theta] = 1 \]

12. Angular Velocity

\[ \omega = \frac{\theta}{t} \]

\[ [\omega] = \frac{1}{T} = T^{-1} \]

13. Angular Acceleration

\[ \alpha = \frac{\omega}{t} \]

\[ [\alpha] = \frac{T^{-1}}{T} = T^{-2} \]

14. Torque

\[ \tau = rF \]

\[ [\tau] = L(MLT^{-2}) = ML^{2}T^{-2} \]

15. Angular Momentum

\[ L = mvr \]

\[ [L] = (M)(LT^{-1})(L) = ML^{2}T^{-1} \]

16. Moment of Inertia

\[ I = mr^{2} \]

\[ [I] = M(L^{2}) = ML^{2} \]

17. Young's Modulus

\[ \text{stress} = \frac{F}{A} \]

\[ \text{stress} = ML^{-1}T^{-2} \]

\[ Y = \frac{\text{stress}}{\text{strain}} \]

\[ [Y] = \frac{ML^{-1}T^{-2}}{1} = ML^{-1}T^{-2} \]

18. Coefficient of Viscosity

\[ \eta = \frac{F}{A} \cdot \frac{dx}{dv} \]

\[ [\eta] = (ML^{-1}T^{-2})(T) = ML^{-1}T^{-1} \]

---

II. Electrical Quantities

1. Electric Charge

\[ Q = It \]

\[ [Q] = I(T) = IT \]

2. Electric Current

\[ I = \frac{Q}{t} \]

\[ [I] = \frac{IT}{T} = I \]

3. Voltage / Potential Difference

\[ V = \frac{W}{Q} \]

\[ [V] = \frac{ML^{2}T^{-2}}{IT} = ML^{2}T^{-3}I^{-1} \]

4. Electric Field

\[ E = \frac{F}{Q} \]

\[ [E] = \frac{MLT^{-2}}{IT} = MLT^{-3}I^{-1} \]

5. Electric Potential

\[ \phi = \frac{W}{Q} \]

\[ [\phi] = \frac{ML^{2}T^{-2}}{IT} = ML^{2}T^{-3}I^{-1} \]

6. Capacitance

\[ C = \frac{Q}{V} \]

\[ [C] = \frac{IT}{ML^{2}T^{-3}I^{-1}} = M^{-1}L^{-2}T^{4}I^{2} \]

7. Resistance

\[ R = \frac{V}{I} \]

\[ [R] = \frac{ML^{2}T^{-3}I^{-1}}{I} = ML^{2}T^{-3}I^{-2} \]

8. Resistivity

\[ \rho = R \frac{A}{l} \]

\[ [\rho] = (ML^{2}T^{-3}I^{-2})(L^{2})(L^{-1}) \]

\[ [\rho] = ML^{3}T^{-3}I^{-2} \]

9. Conductance

\[ G = \frac{1}{R} \]

\[ [G] = (ML^{2}T^{-3}I^{-2})^{-1} = M^{-1}L^{-2}T^{3}I^{2} \]

10. Electrical Power

\[ P = VI \]

\[ [P] = (ML^{2}T^{-3}I^{-1})(I) = ML^{2}T^{-3} \]

11. Electric Dipole Moment

\[ p = qd \]

\[ [p] = ITL \]

---

III. Magnetic Quantities

1. Magnetic Field (B)

\[ F = qvB \]

\[ [B] = \frac{F}{qv} = \frac{MLT^{-2}}{(IT)(LT^{-1})} = MT^{-2}I^{-1} \]

2. Magnetic Flux

\[ \Phi = BA \]

\[ [\Phi] = (MT^{-2}I^{-1})(L^{2}) = ML^{2}T^{-2}I^{-1} \]

3. Magnetic Moment

\[ m = IA \]

\[ [m] = (I)(L^{2}) = L^{2}I \]

4. Inductance

\[ U = \frac{1}{2}LI^{2} \]

\[ [L] = \frac{U}{I^{2}} = \frac{ML^{2}T^{-2}}{I^{2}} = ML^{2}T^{-2}I^{-2} \]

5. Permeability

\[ B = \mu H \]

\[ [\mu] = \frac{B}{H} = \frac{MT^{-2}I^{-1}}{L^{-1}I} = MT^{-2}I^{-2} \]

6. Magnetizing Field (H)

\[ H = \frac{I}{l} \]

\[ [H] = \frac{I}{L} = L^{-1}I \]

7. Magnetic Dipole Moment

\[ M = IA \]

\[ [M] = IL^{2} \]

---

IV. Thermal Quantities

1. Temperature

\[ [\Theta] = \Theta \]

2. Heat energy

\[ Q = mc\Delta T \]

\[ [Q] = (M)(L^{2}T^{-2}\Theta^{-1})(\Theta) = ML^{2}T^{-2} \]

3. Specific Heat

\[ c = \frac{Q}{m\Delta T} \]

\[ [c] = \frac{ML^{2}T^{-2}}{M\Theta} = L^{2}T^{-2}\Theta^{-1} \]

4. Thermal Conductivity

\[ Q = KA \frac{\Delta T}{L} t \]

\[ [K] = \frac{Q\,L}{A\,t\,\Delta T} = \frac{ML^{2}T^{-2} \cdot L}{L^{2}\cdot T \cdot \Theta} \]

\[ [K] = MLT^{-3}\Theta^{-1} \]

5. Coefficient of Expansion

\[ \alpha = \frac{\Delta L}{L\,\Delta T} \]

\[ [\alpha] = \Theta^{-1} \]

---

V. Wave and Oscillation Quantities

1. Frequency

\[ \nu = \frac{1}{T} \]

\[ [\nu] = T^{-1} \]

2. Wavelength

\[ \lambda = \frac{\text{wave velocity}}{\text{frequency}} = \text{wave velocity} \times \text{Time period} \]

\[ \lambda = v \times T \]

\[ [\lambda] = (L T^{-1})(T) = L \]

3. Wave Speed

\[ v = \nu \lambda \]

\[ [v] = (T^{-1})(L) = LT^{-1} \]

4. Spring Constant

\[ F = kx \]

\[ [k] = \frac{F}{x} = \frac{MLT^{-2}}{L} = MT^{-2} \]

5. Time Period

\[ T = \frac{1}{\nu} \]

\[ [T] = T \]

---

VI. Optics Quantities

1. Refractive Index

\[ n = \frac{c}{v} \]

\[ [n] = 1 \]

2. Focal Length

\[ \frac{1}{f} = P \]

\[ [f] = L \]

3. Power of Lens

\[ P = \frac{1}{f} \]

\[ [P] = L^{-1} \]

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VII. Modern Physics Quantities

1. Planck's Constant

\[ E = h\nu \]

\[ [h] = \frac{ML^{2}T^{-2}}{T^{-1}} = ML^{2}T^{-1} \]

2. Work Function

\[ \phi = h\nu_{0} \]

\[ [\phi] = ML^{2}T^{-2} \]

3. Activity

\[ A = \frac{dN}{dt} \]

\[ [A] = T^{-1} \]

4. Wavelength

\[ \lambda = \frac{c}{\nu} \]

\[ [\lambda] = \frac{LT^{-1}}{T^{-1}} = L \]

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More JEE tutorials:

1. Units and measurements class 11 notes

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