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^Limitations of dimensions

December 23, 2021/in Back end /by kp-web-admin

^Limitations of dimensions

Dimensional analysis does not give any information about the scalar or vector nature of the quantity.
Dimensions for v = u + at etc. can’t be calculated in exact form.
Dimensional formulae of the quantities depending on more than three variables cannot be obtained.
Dimensional formulae of the quantities involving trigonometric logarithmic or exponential functions cannot be calculated.
Value of proportionality constant can’t be calculated.

It doesn’t distinguish between the physical quantities having same dimensions.

^Uses of dimensions

December 23, 2021/in Back end /by kp-web-admin

^Uses of dimensions

Correctness of a physical relation can be checked dimensionally. If dimensions of both the sides are found to be same, then the relation is said to be correct otherwise incorrect.
Dimension of unknowns can be calculated.
A mathematical relation between two or three quantities (called a physical relation) can be derived.
Two different system of units can be interrelated.

^Parallel plate capacitor

December 23, 2021/in Back end /by kp-web-admin

Parallel plate capacitor

If the plates X & Y have equal & opposite charges, then charge distribution on plates is as shown in the adjacent drawing. Usually for a parallel plate capacitor we consider this picture.

Let A is the area of each plate, then the charge density of face A, b, c & d are,

Elec. field on the left side (L) of plates, right side of plate (R) & between the plates due to the charge on face a, b, c & d is

Outside the two plates (i.e. in the regions L & R) the electric field due to the charge on the faces a, b, c & d will cancel out, there will no net electric field outside the plates.

Net elec. field between the plates will be

i.e. electric field due to a capacitor plates of equal & opposite charge is zero outside it & inside is which is uniform both in magnitude & direction, (neglecting fringing). Due to this field there will be attractive force between plates, on each plate it will be,

Due to this electric field the potential difference between the plates X & Y separated by distance d is,

Capacitance of a PPC is

If the entire space is filled with a dielectric, then ε₀ → Kε₀ & capacitance becomes

*A quantity having dimensions must posses some units, where as dimensionless quantity can have units. e.g. angle.

December 23, 2021/in Back end /by kp-web-admin

Facts

*A quantity having dimensions must posses some units, where as dimensionless quantity can have units. e.g. angle.

Relative velocity have same dimensions as that of velocity where as relative density is dimensionless.

^Dimensionless Quantities

December 23, 2021/in Back end /by kp-web-admin

^Dimensionless Quantities

Strain, angle, solid angle, all Trigonometric ratios, all real numbers, logarithmic functions, exponential functions, Poisson’s ratio, refractive index, relative density, relative permittivity, relative permeability, fine structure constant etc. have no dimensions.

*Dimensional Variables

December 23, 2021/in Back end /by kp-web-admin

*Dimensional Variables

Are the physical quantities which have dimensions as well as variable values.

e.g. Force, Mass, Velocity, Area, Volume etc.

^Dimensionless Variables

December 23, 2021/in Back end /by kp-web-admin

^Dimensionless Variables

Are the physical quantities which have no dimensions but have variable values.

e.g. Angle, Strain, Specific gravity etc.

*Dimensional Constants

December 23, 2021/in Back end /by kp-web-admin

*Dimensional Constants

All constants are not dimensionless. The physical quantities which have constant values and dimensions as well are called dimensional constants.

e.g. Planck’s constant, Boltzmann’s constant & Gravitational constant etc.

^Dimensionless Constants

December 23, 2021/in Back end /by kp-web-admin

^Dimensionless Constants

Are the physical quantities which have neither dimensions nor variable values.

e.g. real numbers, e, π etc.

^Parallel plates with different charges

December 23, 2021/in Back end /by kp-web-admin

Parallel plates with different charges

If the two parallel metal plates X & Y having charge q₁ & q₂ are placed close to each other. Let the medium between the plates is air or vacuum. then in order to make net electric field in each plate zero, charges redistribute such that inner faces have equal & opposite charges & extreme faces have equal & of same sign as shown below

Elec. field on the left side (L) of plates, right side of plate (R) & between the plates due to the charge on face a, b, c & d is

^Limitations of dimensions

^Uses of dimensions

^Parallel plate capacitor

*A quantity having dimensions must posses some units, where as dimensionless quantity can have units. e.g. angle.

^Dimensionless Quantities

*Dimensional Variables

^Dimensionless Variables

*Dimensional Constants

^Dimensionless Constants

^Parallel plates with different charges

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