Physics
Units and Dimensions
Mole
11
⚡ Quick Summary
A mole is like a 'counting unit' for really tiny things like atoms or molecules. It's a huge number (6.022 x 10^23) that tells you how many of those tiny things are in one mole of a substance.
N/A
One mole of a substance contains exactly 6.02214076 10^23 elementary entities. This number is the fixed numerical value of the Avogadro constant Nₐ when expressed in the unit mol⁻¹ and is called Avogadro number.
Candela
11
⚡ Quick Summary
Candela is a measure of how bright something is *to our eyes*. It's the SI unit for luminous intensity.
N/A
1 candela is the luminous intensity that makes the luminous efficacy of monochromatic radiation of frequency 540 10¹² Hz, Kcd to be 683 when expressed in the unit lm⋅W⁻¹ which is equal to cd⋅sr⋅kg⁻¹m²s³, where kilogram, metre and second are defined in terms of h, c and ΔνCs.
Dimensions of Physical Quantities
11
⚡ Quick Summary
Dimensions tell you what *kind* of quantity you're dealing with (like mass, length, or time), regardless of the units you use to measure it. It's like saying whether you're measuring something's weight versus its height.
N/A
When a quantity is expressed in terms of the base quantities, it is written as a product of different powers of the base quantities. The exponent of a base quantity that enters into the expression is called the dimension of the quantity in that base. Base quantities are represented by one letter symbols: mass (M), length (L), time (T), electric current (I), thermodynamic temperature (K), amount of substance (mol), and luminous intensity (cd).
Dimensional Formula
11
⚡ Quick Summary
A dimensional formula is a short way to show what base quantities (mass, length, time, etc.) make up a physical quantity. It uses M, L, T, etc., with exponents to show how much of each base quantity is involved. For example, the dimensional formula of force is MLT⁻² meaning it depends on mass to the power of 1, length to the power of 1, and time to the power of -2.
[force] = MLT⁻²
An expression for a physical quantity in terms of the base quantities is called the dimensional formula. For example, the dimensional formula of force is MLT⁻².
Uses of Dimensions: Homogeneity of Dimensions
11
⚡ Quick Summary
Imagine an equation as a balanced scale. To keep it balanced, the *kind* of stuff on each side (the dimensions) must be the same. You can't add apples and oranges; similarly, you can't add quantities with different dimensions in a physical equation. This helps you check if your equation makes sense.
N/A
The dimensions of all the terms in an equation must be identical. This principle is called the principle of homogeneity of dimensions. If the dimensions of all the terms are not the same, the equation must be wrong.
Limitations of Dimensional Analysis
11
⚡ Quick Summary
Dimensional analysis is a cool trick, but it has limits! It can't handle dimensionless constants (like π), and it struggles when a quantity depends on more than three other quantities if you're only looking at length, mass, and time. Also, it only estimates!
None mentioned
- It cannot deduce numerical constants having no dimensions.
- It works only if there are as many equations available as there are unknowns. In mechanics, only three base quantities (length, mass, and time) enter. If a quantity depends on more than three quantities, the exponents cannot be uniquely determined.
Order of Magnitude
11
⚡ Quick Summary
Order of magnitude is a quick way to estimate how big or small something is by rounding it to the nearest power of 10. It helps us compare really big and really tiny numbers without getting bogged down in the details.
None mentioned
- Each number is expressed as a ⋅ 10b where 1 ≤ a < 10 and b is an integer.
- To get an approximate idea, round 'a' to 1 if it's ≤ 5, and to 10 if it's > 5. Then the number is approximately 10b.
- The exponent 'b' is called the order of magnitude.
Dimensional Formulae
11
⚡ Quick Summary
A dimensional formula shows how a physical quantity relates to fundamental quantities like Mass (M), Length (L), Time (T), Current (I), and Temperature (K). It's like a recipe showing what basic ingredients make up the quantity.
Examples:
[G] = M⁻¹L³T⁻² (Universal Gravitational Constant)
[S] = MT⁻² (Surface Tension)
[k] = MLT⁻³K⁻¹ (Thermal Conductivity)
[η] = ML⁻¹T⁻¹ (Coefficient of Viscosity)
[Q] = IT (Charge)
[V] = ML²I⁻¹T⁻³ (Potential)
[C] = M⁻¹L⁻²I²T⁴ (Capacitance)
[R] = ML²I⁻²T⁻³ (Resistance)
Dimensional formulas are expressed using square brackets, for example, [Velocity] = LT⁻¹. To find the dimensional formula, express the quantity in terms of fundamental quantities. Remember that constants don't have dimensions.
Units Conversion Using Dimensions
11
⚡ Quick Summary
You can use the dimensions of a physical quantity to convert its value from one unit system (like SI) to another (like CGS). The key is to know the relationship between the base units in each system.
1 joule = 10⁷ erg
General approach: Convert each base unit (M, L, T, etc.) separately using the conversion factors between the systems. Then, combine these factors according to the dimensions of the quantity.
If you know the dimensional formula of a quantity, you can easily convert between unit systems. For example, Energy has dimensions ML²T⁻². 1 Joule (SI unit) equals 10⁷ ergs (CGS unit). This is because 1 kg = 1000 g and 1 m = 100 cm.
Finding Dimensions with Different Basic Quantities
11
⚡ Quick Summary
Instead of using Mass, Length, and Time as the base units, you can choose other quantities like Velocity, Time, and Force as the base. This will change the dimensional formula of other quantities like mass.
[mass] = FT V⁻¹ (when Force, Time, and Velocity are chosen as basic quantities)
You can express all other physical quantities in terms of these new base quantities. The relationship remains based on fundamental physics principles. For instance, if Force, Time, and Velocity are fundamental, Mass can be derived from Force = Mass x Acceleration.
Dimensional Analysis - Checking Equation Correctness
Class 11
⚡ Quick Summary
You can only add or equate things that have the same dimensions. Like, you can't add meters to seconds! If an equation has terms with different dimensions, it's definitely wrong. If all the terms have the same dimensions, it *might* be right.
N/A
The principle of homogeneity states that an equation is dimensionally correct if the dimensions of all the terms in the equation are the same.
Dimensional Analysis - Finding Dimensions of Unknowns
Class 11
⚡ Quick Summary
If you have an equation with some mystery variables, and you know the dimensions of everything else, you can figure out the dimensions of the mystery variables by making sure both sides of the equation have the same dimensions.
N/A
Each term in a physical equation must have the same dimensions. By equating the dimensions of different terms, we can determine the dimensions of unknown quantities.
Dimensional Analysis - Guessing Formulas
Class 11
⚡ Quick Summary
Sometimes, if you know what factors influence something (like force depending on mass, velocity, and radius), you can *guess* the formula by figuring out what powers of each factor would give you the correct dimensions on both sides of the equation. It won't give you any dimensionless constants, though!
F = k h^a r^b v^c (where k is a dimensionless constant and a, b, and c are determined by dimensional analysis)
If a physical quantity depends on other quantities, we can express the relationship as a product of powers of these quantities. By equating the dimensions on both sides, we can find the values of the exponents.
Dimensions of Physical Quantities
11
⚡ Quick Summary
Dimensions tell us what kind of physical quantities we're dealing with (like length, mass, time) and how they relate to each other. It's like the recipe for a physical quantity!
Area: [L<sup>2</sup>], Velocity: [LT<sup>-1</sup>], Force: [MLT<sup>-2</sup>], Energy: [ML<sup>2</sup>T<sup>-2</sup>]
Dimensions are the powers to which fundamental quantities like Mass (M), Length (L), Time (T), Current (I), Temperature (K) etc. are raised to represent a physical quantity. They are independent of the system of units. For example, the dimensions of area are L2, and the dimensions of velocity are LT-1.
Dimensional Analysis
11
⚡ Quick Summary
Dimensional analysis is a tool to check if an equation is correct or to find the relationship between different physical quantities. Think of it as a mathematical spell-checker for physics!
Principle of Homogeneity: [Left Hand Side] = [Right Hand Side]
Dimensional analysis involves checking the consistency of equations by ensuring that the dimensions on both sides of the equation are the same. It can also be used to derive relationships between physical quantities if the dependencies are known, up to a dimensionless constant.
Planck's Constant
11
⚡ Quick Summary
Planck's constant (h) relates the energy of a photon to its frequency. It's a tiny but important number in quantum mechanics. It tells us how energy is quantized (comes in packets).
E = hν => [h] = [E]/[ν] = ML<sup>2</sup>T<sup>-2</sup> / T<sup>-1</sup> = ML<sup>2</sup>T<sup>-1</sup>
Planck's constant (h) appears in the equation E = hν, where E is energy and ν is frequency. The dimension of Planck's constant can be derived from this equation.
Ohm's Law
11
⚡ Quick Summary
Ohm's law relates voltage, current, and resistance in a circuit. Voltage is like water pressure, current is like the flow rate, and resistance is like the size of the pipe. If you know two, you can find the third.
V = IR
Ohm's law states that the potential difference (V) across a conductor is proportional to the current (I) flowing through it, with the constant of proportionality being the resistance (R). Using dimensional analysis and knowing the dimensions of V, I, and R, we can verify or derive Ohm's Law.
Kinetic Energy of Rotating Body
11
⚡ Quick Summary
A rotating body has energy due to its rotation, called kinetic energy. It depends on how heavy it is (moment of inertia) and how fast it's spinning (angular speed).
K = k I<sup>a</sup>ω<sup>b</sup>
The kinetic energy (K) of a rotating body depends on its moment of inertia (I) and angular speed (ω). The relationship can be expressed as K = k Iaωb, where k is a dimensionless constant. By using dimensional analysis, we can find the values of a and b.
Mass-Energy Equivalence
11
⚡ Quick Summary
Einstein's famous equation, E=mc², says that mass and energy are interchangeable. A tiny bit of mass can be converted into a huge amount of energy, and vice-versa. This is how nuclear power works!
E = mc<sup>2</sup>
Einstein's theory of relativity shows that mass (m) can be converted into energy (E). The relationship between energy, mass, and the speed of light (c) can be found by dimensional analysis.
Frequency of Vibration of a String
11
⚡ Quick Summary
The frequency at which a string vibrates depends on its length, how tight it is (tension), and how heavy it is per unit length. Think of a guitar string – shorter, tighter, and lighter strings vibrate faster and produce higher notes.
ν ∝ (1/L) * √(F/m)
The frequency (ν) of vibration of a string depends on the length (L) between the nodes, the tension (F) in the string, and its mass per unit length (m). Dimensional analysis can be used to find the expression for the frequency.
Physical Quantities, Units, and Dimensions
11/Higher Secondary
⚡ Quick Summary
This section provides a table listing common physical quantities, their standard SI units, and their corresponding dimensions in terms of fundamental quantities like Mass (M), Length (L), Time (T), Current (I), and Temperature (Θ). It's a reference for understanding the relationship between different physical quantities.
This section primarily presents a reference table, not formulas for calculation. However, the DIMENSIONS of each quantity can be used in dimensional analysis to derive or check formulas.
Physical Quantities, SI Units, and Dimensions
This appendix provides a comprehensive list of physical quantities, their corresponding SI units, and their dimensional formulas. Dimensions are expressed in terms of fundamental quantities: Mass (M), Length (L), Time (T), Electric Current (I), and Thermodynamic Temperature (Θ).
Key Concepts:
- Physical Quantity: A measurable aspect of the physical world.
- SI Unit: The standard unit of measurement defined by the International System of Units.
- Dimension: The fundamental nature of a physical quantity, expressed in terms of fundamental quantities (M, L, T, I, Θ). Dimensional analysis is a powerful tool for checking the consistency of equations and deriving relationships between physical quantities.
Table of Physical Quantities, SI Units, and Dimensions:
| Quantity | Common Symbol | SI Unit | Dimension |
|---|---|---|---|
| Displacement | s | metre (m) | L |
| Mass | m, M | kilogram (kg) | M |
| Time | t | second (s) | T |
| Area | A | m2 | L2 |
| Volume | V | m3 | L3 |
| Density | ρ | kg m-3 | ML-3 |
| Velocity | v, u | m s-1 | LT-1 |
| Acceleration | a | m s-2 | LT-2 |
| Force | F | newton (N) | MLT-2 |
| Work | W | joule (J) (= N–m) | ML2T-2 |
| Energy | E, U, K | joule (J) | ML2T-2 |
| Power | P | watt (W) (= J s-1) | ML2T-3 |
| Momentum | p | kg–m s-1 | MLT-1 |
| Gravitational constant | G | N–m2 kg-2 | L3M-1T-2 |
| Angle | θ, φ | radian | (Dimensionless) |
| Angular velocity | ω | rad s-1 | T-1 |
| Angular acceleration | α | rad s-2 | T-2 |
| Angular momentum | L | kg–m2 s-1 | ML2T-1 |
| Moment of inertia | I | kg–m2 | ML2 |
| Torque | τ | N–m | ML2T-2 |
| Angular frequency | ω | rad s-1 | T-1 |
| Frequency | ν | hertz (Hz) | T-1 |
| Period | T | s | T |
| Young’s modulus | Y | N m-2 | ML-1T-2 |
| Bulk modulus | B | N m-2 | ML-1T-2 |
| Shear modulus | η | N m-2 | ML-1T-2 |
| Surface tension | S | N m-1 | MT-2 |
| Coefficient of viscosity | η | N–s m-2 | ML-1T-1 |
| Pressure | P, p | N m-2, Pa | ML-1T-2 |
| Wavelength | λ | m | L |
| Intensity of wave | I | W m-2 | MT-3 |
| Temperature | T | kelvin (K) | Θ |
| Specific heat capacity | c | J kg-1–K-1 | L2T-2Θ-1 |
| Stefan’s constant | σ | W m-2–K-4 | MT-3Θ-4 |
| Heat | Q | J | ML2T-2 |
| Thermal conductivity | K | W m-1–K-1 | MLT-3Θ-1 |
| Current | I | ampere (A) | I |
| Charge | q, Q | coulomb (C) | IT |
| Current density | j | A m-2 | IL-2 |
| Electrical conductivity | σ | 1/Ω–m (= mho/m) | I2T3M-1L-3 |
| Dielectric constant | k | (Dimensionless) | (Dimensionless) |
| Electric dipole moment | p | C–m | LIT |
| Electric field | E | V m-1 (= N C-1) | MLI-1T-3 |
| Potential (voltage) | V | volt (V) (= J C-1) | ML2I-1T-3 |
| Electric flux | Φ | V–m | ML3I-1T-3 |
| Capacitance | C | farad (F) | I2T4M-1L-2 |
| Electromotive force | E | volt (V) | ML2I-1T-3 |
| Resistance | R | ohm (Ω) | ML2I-2T-3 |
| Permittivity of space | ε0 | C2 N-1–m-2 (= F m-1) | I2T4M-1L-3 |
| Permeability of space | μ0 | N A-2 | MLI-2T-2 |
| Magnetic field | B | tesla (T) (= Wb m-2) | MI-1T-2 |
| Magnetic flux | ΦB | weber (Wb) | ML2I-1T-2 |
| Magnetic dipole moment | μ | N–m T-1 | IL2 |
| Inductance | L | henry (H) | ML2I-2T-2 |
Dimensions
11
⚡ Quick Summary
The fundamental physical quantities that make up a unit.
N/A - Conceptual understanding of representing physical quantities in terms of mass (M), length (L), and time (T).
Explains the use of dimensions to check the consistency of equations and derive relationships between physical quantities.