CGS–SI Conversions and Dimensions (JEE Mains)

CGS–SI Conversions and Dimensions (JEE Mains)

Use: Quick reference for Units & Measurements, Mechanics, Properties of Matter, Electricity, Magnetism and Modern Physics.

1. SI Base Quantities

Quantity

SI unit

SI symbol

Dimension

Length

metre

m

[L]

Mass

kilogram

kg

[M]

Time

second

s

[T]

Electric current

ampere

A

[I]

Temperature

kelvin

K

[Θ]

Amount of substance

mole

mol

[N]

Luminous intensity

candela

cd

[J]

2. CGS Base Units

Quantity

CGS unit

Symbol

Length

centimetre

cm

Mass

gram

g

Time

second

s

Force

dyne

dyn

Energy/work

erg

erg

Pressure

barye

Ba

Power

erg s⁻¹

—

Viscosity

poise

P

Kinematic viscosity

stoke

St

Surface tension

dyn cm⁻¹

—

3. Fundamental Conversions

  • 1 m = 10² cm.

  • 1 km = 10⁵ cm.

  • 1 kg = 10³ g.

  • 1 s = 1 s.

  • 1 N = 10⁵ dyn.

  • 1 J = 10⁷ erg.

  • 1 Pa = 10 Ba = 10 dyn cm⁻².

  • 1 W = 10⁷ erg s⁻¹.

  • 1 m² = 10⁴ cm².

  • 1 m³ = 10⁶ cm³.

  • 1 L = 10³ cm³ = 10⁻³ m³.

  • 1 mL = 1 cm³.

  • 1 kg m⁻³ = 10⁻³ g cm⁻³.

  • 1 g cm⁻³ = 10³ kg m⁻³.

General conversion rule

If a quantity has dimensions [MᵃLᵇTᶜIᵈΘᵉ], then its numerical value changes according to the conversion factors for each base quantity.

For CGS to SI:

  • M: 1 g = 10⁻³ kg.

  • L: 1 cm = 10⁻² m.

  • T: unchanged.

  • I: unchanged in SI electrical quantities, but electromagnetic CGS systems require care; use the specific conversion table below.

4. Mechanics: SI, CGS and Dimensions

Quantity

Formula

SI unit

CGS unit

SI dimensions

Area

l²

m²

cm²

[L²]

Volume

l³

m³

cm³

[L³]

Density

m/V

kg m⁻³

g cm⁻³

[ML⁻³]

Velocity

dx/dt

m s⁻¹

cm s⁻¹

[LT⁻¹]

Acceleration

dv/dt

m s⁻²

cm s⁻²

[LT⁻²]

Momentum

mv

kg m s⁻¹

g cm s⁻¹

[MLT⁻¹]

Force

ma

N

dyn

[MLT⁻²]

Impulse

FΔt

N s

dyn s

[MLT⁻¹]

Work/energy

Fs

J

erg

[ML²T⁻²]

Power

W/t

W

erg s⁻¹

[ML²T⁻³]

Pressure/stress

F/A

Pa

Ba = dyn cm⁻²

[ML⁻¹T⁻²]

Strain

Δl/l

dimensionless

dimensionless

[1]

Young’s modulus

stress/strain

Pa

dyn cm⁻²

[ML⁻¹T⁻²]

Bulk modulus

pressure/(ΔV/V)

Pa

dyn cm⁻²

[ML⁻¹T⁻²]

Shear modulus

shear stress/shear strain

Pa

dyn cm⁻²

[ML⁻¹T⁻²]

Spring constant

F/x

N m⁻¹

dyn cm⁻¹

[MT⁻²]

Surface tension

F/l

N m⁻¹

dyn cm⁻¹

[MT⁻²]

Coefficient of friction

friction/normal force

dimensionless

dimensionless

[1]

Angular velocity

dθ/dt

rad s⁻¹

rad s⁻¹

[T⁻¹]

Angular acceleration

dω/dt

rad s⁻²

rad s⁻²

[T⁻²]

Torque

rF

N m

dyn cm

[ML²T⁻²]

Angular momentum

r × p

kg m² s⁻¹

g cm² s⁻¹

[ML²T⁻¹]

Moment of inertia

mr²

kg m²

g cm²

[ML²]

Frequency

1/T

Hz

s⁻¹

[T⁻¹]

Gravitational constant

Fr²/m²

N m² kg⁻²

dyn cm² g⁻²

[M⁻¹L³T⁻²]

Acceleration due to gravity

F/m

m s⁻²

cm s⁻²

[LT⁻²]

Coefficient of restitution

relative speed ratio

dimensionless

dimensionless

[1]

Action

energy × time

J s

erg s

[ML²T⁻¹]

Important conversion examples

  • 1 N = 1 kg m s⁻² = 10⁵ g cm s⁻² = 10⁵ dyn.

  • 1 J = 1 N m = 10⁵ dyn × 10² cm = 10⁷ erg.

  • 1 Pa = 1 N m⁻² = 10⁵ dyn/10⁴ cm² = 10 dyn cm⁻².

  • 1 N m = 10⁷ dyn cm.

  • 1 kg m² = 10⁷ g cm².

5. Fluid Mechanics and Thermal Physics

Quantity

Formula

SI unit

CGS unit

Dimensions

Dynamic viscosity

shear stress/velocity gradient

Pa s

poise (P)

[ML⁻¹T⁻¹]

Kinematic viscosity

η/ρ

m² s⁻¹

stoke (St)

[L²T⁻¹]

Surface tension

F/l

N m⁻¹

dyn cm⁻¹

[MT⁻²]

Pressure

F/A

Pa

Ba

[ML⁻¹T⁻²]

Volume flow rate

V/t

m³ s⁻¹

cm³ s⁻¹

[L³T⁻¹]

Mass flow rate

m/t

kg s⁻¹

g s⁻¹

[MT⁻¹]

Heat

energy

J

erg

[ML²T⁻²]

Specific heat

heat/(mass × temperature)

J kg⁻¹ K⁻¹

erg g⁻¹ K⁻¹

[L²T⁻²Θ⁻¹]

Heat capacity

heat/temperature

J K⁻¹

erg K⁻¹

[ML²T⁻²Θ⁻¹]

Specific latent heat

heat/mass

J kg⁻¹

erg g⁻¹

[L²T⁻²]

Thermal conductivity

heat × length/(area × time × temperature)

W m⁻¹ K⁻¹

erg cm⁻¹ s⁻¹ K⁻¹

[MLT⁻³Θ⁻¹]

Entropy

heat/temperature

J K⁻¹

erg K⁻¹

[ML²T⁻²Θ⁻¹]

Gas constant R

PV/(nT)

J mol⁻¹ K⁻¹

erg mol⁻¹ K⁻¹

[ML²T⁻²Θ⁻¹N⁻¹]

Boltzmann constant k_B

energy/temperature

J K⁻¹

erg K⁻¹

[ML²T⁻²Θ⁻¹]

Stefan constant σ

power/(area × T⁴)

W m⁻² K⁻⁴

erg cm⁻² s⁻¹ K⁻⁴

[MT⁻³Θ⁻⁴]

Wien constant b

wavelength × temperature

m K

cm K

[LΘ]

Viscosity conversions

  • 1 poise = 0.1 Pa s.

  • 1 Pa s = 10 poise.

  • 1 centipoise = 10⁻³ Pa s.

  • 1 stoke = 10⁻⁴ m² s⁻¹.

  • 1 m² s⁻¹ = 10⁴ stoke.

  • 1 poise = 1 g cm⁻¹ s⁻¹.

Surface tension conversion

  • 1 dyn cm⁻¹ = 10⁻³ N m⁻¹.

  • 1 N m⁻¹ = 10³ dyn cm⁻¹.

6. Electricity: SI and Dimensions

Quantity

Formula

SI unit

CGS/other common unit

Dimensions

Charge

It

coulomb (C)

statcoulomb/esu in electrostatic CGS

[IT]

Current

q/t

ampere

abampere in electromagnetic CGS

[I]

Potential difference

W/q

volt

statvolt in esu

[ML²T⁻³I⁻¹]

Electric field

F/q

N C⁻¹ or V m⁻¹

dyn esu⁻¹

[MLT⁻³I⁻¹]

Electric flux

E·A

N m² C⁻¹

—

[ML³T⁻³I⁻¹]

Permittivity

ε₀

F m⁻¹

—

[M⁻¹L⁻³T⁴I²]

Capacitance

q/V

farad (F)

—

[M⁻¹L⁻²T⁴I²]

Resistance

V/I

ohm (Ω)

—

[ML²T⁻³I⁻²]

Resistivity

RA/l

Ω m

Ω cm

[ML³T⁻³I⁻²]

Conductance

1/R

siemens (S)

—

[M⁻¹L⁻²T³I²]

Conductivity

1/ρ

S m⁻¹

S cm⁻¹

[M⁻¹L⁻³T³I²]

Current density

I/A

A m⁻²

A cm⁻²

[IL⁻²]

Electric dipole moment

ql

C m

esu cm

[ITL]

Electric susceptibility

P/(ε₀E)

dimensionless

—

[1]

Polarisation

dipole moment/volume

C m⁻²

—

[IL⁻²T]

Energy density

energy/volume

J m⁻³

erg cm⁻³

[ML⁻¹T⁻²]

Key electrical conversions

  • 1 C = 2.998 × 10⁹ statC approximately.

  • 1 statC ≈ 3.336 × 10⁻¹⁰ C.

  • 1 V m⁻¹ = 10⁻² V cm⁻¹.

  • 1 V cm⁻¹ = 100 V m⁻¹.

  • 1 Ω m = 100 Ω cm.

  • 1 Ω cm = 10⁻² Ω m.

  • 1 S m⁻¹ = 10⁻² S cm⁻¹.

  • 1 S cm⁻¹ = 100 S m⁻¹.

  • 1 eV = 1.602 × 10⁻¹⁹ J.

  • 1 electron charge e = 1.602 × 10⁻¹⁹ C.

7. Magnetism: SI and CGS

Magnetic CGS has multiple conventions. For JEE Main, remember the standard practical conversions below and use the unit system explicitly given in the question.

Quantity

SI unit

Common CGS unit

SI dimensions

Magnetic field B

tesla (T)

gauss (G)

[MT⁻²I⁻¹]

Magnetic flux

weber (Wb)

maxwell (Mx)

[ML²T⁻²I⁻¹]

Magnetic field strength H

A m⁻¹

Oersted (Oe)

[IL⁻¹]

Magnetic moment

A m²

emu

[IL²]

Magnetisation

A m⁻¹

emu cm⁻³

[IL⁻¹]

Permeability

H m⁻¹

—

[MLT⁻²I⁻²]

Magnetic susceptibility

dimensionless

dimensionless

[1]

Inductance

henry (H)

—

[ML²T⁻²I⁻²]

Magnetic conversions

  • 1 T = 10⁴ G.

  • 1 G = 10⁻⁴ T.

  • 1 Wb = 10⁸ Mx.

  • 1 Mx = 10⁻⁸ Wb.

  • 1 A m² = 10³ emu.

  • 1 emu = 10⁻³ A m².

  • In vacuum: B = μ₀H in SI.

  • μ₀ = 4π × 10⁻⁷ H m⁻¹ = 4π × 10⁻⁷ N A⁻².

8. Optics and Wave Quantities

Quantity

Formula

SI unit

CGS/common unit

Dimensions

Wavelength

λ

m

cm, Å, nm

[L]

Frequency

ν

Hz

s⁻¹

[T⁻¹]

Wave number

1/λ

m⁻¹

cm⁻¹

[L⁻¹]

Wave speed

νλ

m s⁻¹

cm s⁻¹

[LT⁻¹]

Intensity

power/area

W m⁻²

erg s⁻¹ cm⁻²

[MT⁻³]

Refractive index

c/v

dimensionless

—

[1]

Focal length

f

m

cm

[L]

Lens power

1/f(m)

dioptre (D)

cm⁻¹ if f in cm

[L⁻¹]

Optical path length

μl

m

cm

[L]

Conversions

  • 1 nm = 10⁻⁹ m = 10⁻⁷ cm.

  • 1 Å = 10⁻¹⁰ m = 10⁻⁸ cm.

  • 1 μm = 10⁻⁶ m = 10⁻⁴ cm.

  • 1 cm⁻¹ = 100 m⁻¹.

  • 1 D = 1 m⁻¹ for focal length in metres; numerically P(D) = 100/f(cm).

9. Modern Physics and Nuclear Physics

Quantity

Formula

SI unit

CGS/common unit

Dimensions

Planck constant h

E/ν

J s

erg s

[ML²T⁻¹]

Reduced Planck constant ħ

h/2π

J s

erg s

[ML²T⁻¹]

Photon energy

hν = hc/λ

J

erg

[ML²T⁻²]

Work function

energy

J or eV

erg

[ML²T⁻²]

Electron volt

—

eV

—

energy

Mass defect

Δm

kg

g

[M]

Nuclear energy

Δmc²

J

erg

[ML²T⁻²]

Activity

−dN/dt

Bq

Ci

[T⁻¹]

Decay constant

λ

s⁻¹

s⁻¹

[T⁻¹]

Half-life

0.693/λ

s

s

[T]

Absorbed dose

energy/mass

gray (Gy)

rad

[L²T⁻²]

Equivalent dose

dose × quality factor

sievert (Sv)

rem

[L²T⁻²]

Atomic mass unit

1 u

kg

—

[M]

Conversions

  • 1 u = 1.6605 × 10⁻²⁷ kg = 1.6605 × 10⁻²⁴ g.

  • 1 u c² = 931.5 MeV approximately.

  • 1 MeV = 10⁶ eV.

  • 1 eV = 1.602 × 10⁻¹⁹ J.

  • 1 Ci = 3.7 × 10¹⁰ Bq.

  • 1 Bq = 1 s⁻¹.

  • 1 Gy = 100 rad.

  • 1 Sv = 100 rem.

10. Dimensionless Quantities

The following are dimensionless:

  • Strain.

  • Relative density and relative velocity.

  • Coefficient of friction.

  • Coefficient of restitution.

  • Refractive index.

  • Dielectric constant.

  • Relative permittivity.

  • Relative permeability.

  • Specific gravity.

  • Plane angle in radians.

  • Solid angle in steradians.

  • Mathematical constants π and e.

Important: Dimensionless does not necessarily mean unitless in every practical convention; radians and steradians are treated as dimensionless SI derived units.

11. Derived SI and CGS Units: Must Memorise

Quantity

SI

CGS

Force

1 N = kg m s⁻²

1 dyn = g cm s⁻²

Energy

1 J = kg m² s⁻²

1 erg = g cm² s⁻²

Pressure

1 Pa = N m⁻²

1 Ba = dyn cm⁻²

Power

1 W = J s⁻¹

erg s⁻¹

Viscosity

Pa s

poise

Kinematic viscosity

m² s⁻¹

stoke

Magnetic field

tesla

gauss

Magnetic flux

weber

maxwell

Activity

becquerel

—

Radiation dose

gray

rad

12. JEE Main Conversion Shortcuts

  • For [MᵃLᵇTᶜ], convert g→kg and cm→m directly:

    • g contributes 10⁻³ per mass power.

    • cm contributes 10⁻² per length power.

    • s remains unchanged.

  • Example: 1 g cm⁻¹ s⁻² = 10⁻³ kg × 10² m⁻¹ s⁻² = 0.1 Pa.

  • Example: 1 g cm² s⁻² = 10⁻³ × 10⁻⁴ kg m² s⁻² = 10⁻⁷ J = 1 erg.

  • Example: 1 g cm⁻³ = 10³ kg m⁻³.

  • Always convert area and volume with squared and cubed factors:

    • 1 cm² = 10⁻⁴ m².

    • 1 cm³ = 10⁻⁶ m³.

13. Important JEE Main Lines and Traps

  1. CGS force is dyne and SI force is newton: 1 N = 10⁵ dyn.

  1. CGS energy is erg and SI energy is joule: 1 J = 10⁷ erg.

  1. Pressure conversion: 1 Pa = 10 dyn cm⁻², not 10⁻¹⁰ dyn cm⁻².

  1. Density conversion: 1 g cm⁻³ = 1000 kg m⁻³.

  1. Work and torque have the same dimensions but are not the same physical quantity.

  1. Pressure and energy density have the same dimensions.

  1. In electromagnetic CGS, esu and emu are different systems; do not mix electrostatic and electromagnetic units without a stated conversion.

  1. For electrical quantities, SI is safest in JEE Main unless the question explicitly gives CGS units.

  1. Dimensionless quantities may still carry named units such as rad and sr.

  1. Convert cm² and cm³ using squared and cubed conversion factors, respectively.

  1. A quantity’s numerical value changes when its unit changes; its dimensions do not.

  1. Dimensional analysis cannot determine pure numerical constants such as 2, π or 1/2.

  1. Dimensional correctness is necessary but not sufficient for correctness.

14. One-Page Final Revision

1 m = 10² cm              1 kg = 10³ g
1 N = 10⁵ dyn             1 J = 10⁷ erg
1 Pa = 10 dyn cm⁻²        1 W = 10⁷ erg s⁻¹
1 P = 0.1 Pa s             1 St = 10⁻⁴ m² s⁻¹
1 T = 10⁴ G                1 Wb = 10⁸ Mx
1 eV = 1.602×10⁻¹⁹ J      1 u c² = 931.5 MeV
1 Gy = 100 rad             1 Ci = 3.7×10¹⁰ Bq

Core dimensions

v  [LT⁻¹]          a  [LT⁻²]
p  [MLT⁻¹]         F  [MLT⁻²]
W,E [ML²T⁻²]       P  [ML²T⁻³]
pressure [ML⁻¹T⁻²] ρ [ML⁻³]
G  [M⁻¹L³T⁻²]      h  [ML²T⁻¹]
q  [IT]            V  [ML²T⁻³I⁻¹]
R  [ML²T⁻³I⁻²]     C  [M⁻¹L⁻²T⁴I²]
B  [MT⁻²I⁻¹]      μ₀ [MLT⁻²I⁻²]

Final unit-check workflow

Write formula → substitute dimensions → simplify powers → compare both sides