Understanding the Dimensions of Mobility

Mobility is an important physical quantity in electromagnetism and current electricity. It quantifies how easily charge carriers such as electrons or ions move through a material when an electric field is applied. The dimensions of mobility are commonly asked in JEE Main and other competitive examinations.

Mobility: Definition and Physical Meaning

Mobility, commonly denoted by $\mu$, is defined as the magnitude of the drift velocity acquired by a charge carrier per unit applied electric field. It is a key parameter in understanding electrical conduction in solids, especially in semiconductors and metals. Mobility reflects the effectiveness of charge transport under the influence of an external electric field.

Mathematical Expression and Unit of Mobility

The mobility $\mu$ of a charge carrier is mathematically expressed as $\mu = \dfrac{v_d}{E}$, where $v_d$ is the drift velocity and $E$ is the electric field applied to the material. The SI unit of mobility is $\mathrm{m^2\,V^{-1}\,s^{-1}}$.

Detailed Derivation of the Dimensions of Mobility

To obtain the dimensional formula for mobility, consider the equation $\mu = \dfrac{v_d}{E}$. The drift velocity $v_d$ has the dimensions $[L\,T^{-1}]$, as it represents the average velocity of charge carriers. The electric field $E$ has the dimensions $[M^{1}L^{1}T^{-3}I^{-1}]$ in the SI system, where $I$ represents electric current.

Dividing their respective dimensions,

$\mu = \dfrac{[L\,T^{-1}]}{[M^{1}\,L^{1}\,T^{-3}\,I^{-1}]} = [M^{-1}\,L^{0}\,T^{2}\,I^{1}]$

Thus, the dimensional formula for mobility is $[M^{-1}\,L^{0}\,T^{2}\,I^{1}]$.

Units and Dimensional Formula: Summary Table

Alternative Method Using SI Units

Drift velocity has the SI unit $\mathrm{m\,s^{-1}}$ and the electric field has the SI unit $\mathrm{V\,m^{-1}}$. Therefore, mobility has the unit:

$\mathrm{m\,s^{-1}\,/\,V\,m^{-1} = m^2\,V^{-1}\,s^{-1}}$

Expressing volt in SI base units as $\mathrm{V = kg\,m^2\,A^{-1}\,s^{-3}}$, the dimensions simplify to $[M^{-1}L^{0}T^{2}I^{1}]$.

Mobility in Terms of Physical Quantities

The mobility of charge carriers can also be related to their charge ($e$), relaxation time ($\tau$), and mass ($m$) as $\mu = \dfrac{e\tau}{m}$. However, for dimensional analysis, the primary definition $\mu = \dfrac{v_d}{E}$ is required.

Applications of Dimensions of Mobility

Knowing the dimensional formula of mobility helps verify equations through dimensional analysis and is essential in solving problems based on current conduction in conductors and semiconductors for JEE Main. Mobility is directly used in expressions for electrical conductivity and resistivity.

• Determining semiconductor type using mobility values
• Verifying circuit equations in electromagnetism
• Solving numericals involving drift velocity
• Analyzing dependence of current on carrier properties

Important Related Quantities

The study of mobility is connected with other dimensional quantities such as resistance, density, and electric flux. For reference, the Dimensions Of Resistance and the Dimensions Of Electric Flux are commonly used in electromagnetic theory.

Further concepts, including stress, work, and even the Dimensions Of Density and Dimensions Of Work, rely on dimensional analysis for verifying physics formulas.

A comprehensive understanding of the dimensions of mobility enables students to approach related topics, including Dimensions Of Stress problems and new units like the Dimensions Of Light Year, with greater accuracy for examinations.