Answer
Verified
398.4k+ views
Hint: According to Ohm’s law, V = iR, where V is potential difference, i is current and R is resistance. Hence, $R=\dfrac{V}{i}$. Use this formula to find the dimensional formula of R. V is work done per unit charge.
Formula used:
V = iR
$\left[ W \right]=\left[ M{{L}^{2}}{{T}^{-2}} \right]$
[q] = [AT]
Complete step by step answer:
Electrical resistance of a given substance is the ability of the substance to resist the flow of electrons or charges when a potential difference is created across the substance.
The flow of charges per unit time is called current in the circuit.
Suppose a conductor with a resistance of R is connected across a cell of emf V. This cell will create a potential difference V across the conductor. Due to the potential difference there will be current in the circuit. Let the current in the circuit be i.
According to Ohm’s law, V = iR.
Hence, we get that
$R=\dfrac{V}{i}$.
Hence, the dimensional formula of resistance will be equal to the ratio of dimensional formulas of potential difference to current. $\left[ R \right]=\dfrac{\left[ V \right]}{\left[ i \right]}$ …. (i).
Therefore, let us find the dimensional formula of V and i.
Potential difference is equal to the work done per unit charge.
The dimensional formulas of work done is $\left[ W \right]=\left[ M{{L}^{2}}{{T}^{-2}} \right]$.
The dimensional formula of charge is [q] = [AT].
Hence, the dimensional formula of potential difference is $\left[ V \right]=\dfrac{\left[ W \right]}{\left[ q \right]}=\dfrac{\left[ M{{L}^{2}}{{T}^{-2}} \right]}{\left[ AT \right]}=\left[ M{{L}^{2}}{{T}^{-3}}{{A}^{-1}} \right]$.
The dimensional formula of current is [A].
Substitute the dimensional formulas of potential difference and current in equation (i).
$\left[ R \right]=\dfrac{\left[ M{{L}^{2}}{{T}^{-3}}{{A}^{-1}} \right]}{\left[ A \right]}=\left[ M{{L}^{2}}{{T}^{-3}}{{A}^{-2}} \right]$.
Hence, the correct answer is option B.
Note:
We can also use the units of the quantities for finding the dimensional formula of resistances.
The unit of work is $kg{{m}^{2}}{{s}^{-2}}$.
The unit of charge is As.
Therefore, the unit of potential difference is $kg{{m}^{2}}{{s}^{-3}}{{A}^{-1}}$.
The unit of current is A.
Hence,the unit of resistance is $\dfrac{kg{{m}^{2}}{{s}^{-3}}{{A}^{-1}}}{A}=kg{{m}^{2}}{{s}^{-3}}{{A}^{-2}}$.
The dimensions of units kg, m, s and A are M, L, T and A.
Hence, the dimensions of resistance are $\left[ M{{L}^{2}}{{T}^{-3}}{{A}^{-2}} \right]$.
Formula used:
V = iR
$\left[ W \right]=\left[ M{{L}^{2}}{{T}^{-2}} \right]$
[q] = [AT]
Complete step by step answer:
Electrical resistance of a given substance is the ability of the substance to resist the flow of electrons or charges when a potential difference is created across the substance.
The flow of charges per unit time is called current in the circuit.
Suppose a conductor with a resistance of R is connected across a cell of emf V. This cell will create a potential difference V across the conductor. Due to the potential difference there will be current in the circuit. Let the current in the circuit be i.
According to Ohm’s law, V = iR.
Hence, we get that
$R=\dfrac{V}{i}$.
Hence, the dimensional formula of resistance will be equal to the ratio of dimensional formulas of potential difference to current. $\left[ R \right]=\dfrac{\left[ V \right]}{\left[ i \right]}$ …. (i).
Therefore, let us find the dimensional formula of V and i.
Potential difference is equal to the work done per unit charge.
The dimensional formulas of work done is $\left[ W \right]=\left[ M{{L}^{2}}{{T}^{-2}} \right]$.
The dimensional formula of charge is [q] = [AT].
Hence, the dimensional formula of potential difference is $\left[ V \right]=\dfrac{\left[ W \right]}{\left[ q \right]}=\dfrac{\left[ M{{L}^{2}}{{T}^{-2}} \right]}{\left[ AT \right]}=\left[ M{{L}^{2}}{{T}^{-3}}{{A}^{-1}} \right]$.
The dimensional formula of current is [A].
Substitute the dimensional formulas of potential difference and current in equation (i).
$\left[ R \right]=\dfrac{\left[ M{{L}^{2}}{{T}^{-3}}{{A}^{-1}} \right]}{\left[ A \right]}=\left[ M{{L}^{2}}{{T}^{-3}}{{A}^{-2}} \right]$.
Hence, the correct answer is option B.
Note:
We can also use the units of the quantities for finding the dimensional formula of resistances.
The unit of work is $kg{{m}^{2}}{{s}^{-2}}$.
The unit of charge is As.
Therefore, the unit of potential difference is $kg{{m}^{2}}{{s}^{-3}}{{A}^{-1}}$.
The unit of current is A.
Hence,the unit of resistance is $\dfrac{kg{{m}^{2}}{{s}^{-3}}{{A}^{-1}}}{A}=kg{{m}^{2}}{{s}^{-3}}{{A}^{-2}}$.
The dimensions of units kg, m, s and A are M, L, T and A.
Hence, the dimensions of resistance are $\left[ M{{L}^{2}}{{T}^{-3}}{{A}^{-2}} \right]$.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Summary of the poem Where the Mind is Without Fear class 8 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Write an application to the principal requesting five class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE