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Units and Dimensions class 11

Units and Dimensions notes JEE
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Notes • Solved Problems

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

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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