Units Dimentions Error QA 2
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The document discusses dimensional consistency in physics, specifically focusing on expressions for length scale influenced by various physical constants and quantities. It analyzes which expressions are dimensionally correct based on the definitions of mass, length, time, temperature, and electric current. The correct options identified from the analysis are (b) and (d).
![Physics Helpline
L K Satapathy
Dimensional Consistency
Units Dimensions Error 2
[ Mass ] = [ M ]
[ Length ] = [ L ]
[ Time ] = [ T ]
[ Electric Current ] = [ A ]
[ Temperature ] = [ K ]](https://image.slidesharecdn.com/ude2-160608053437/85/Units-Dimentions-Error-QA-2-1-320.jpg)
![Physics Helpline
L K Satapathy Units Dimensions Error 2
Question : A length scale ( l ) , depends on the permittivity () of a dielectric
material , Boltzmann constant ( kB ) , the absolute temperature ( T ) , the number
per unit volume (n) of certain charged particles and the charge (q) carried by each
of the particles . Which of the following expression(s) for l is(are) dimensionally
correct ?
2 2 2
2 2 3 1 3
( ) ( ) ( ) ( )B
B B B
k Tnq q q
a l b l c l d l
k T nq n k T n k T
Answer :
We will use the following nomenclature for
the dimensions of fundamental quantities
[ mass ] = [ M ]
[ length ] = [ L ]
[ time ] = [ T ]
[ temperature ] = [ K ]
[ electric current ] = [ A ]](https://image.slidesharecdn.com/ude2-160608053437/85/Units-Dimentions-Error-QA-2-2-320.jpg)
![Physics Helpline
L K Satapathy Units Dimensions Error 2
1 2 2
1 2 2 1
1
[ ]
[ ] [ ]
[ ]
B
energy M L T
k M L T K
temp K
3
[ ] [ ]n L
(1) Electrostatic force between two point charges
2 2 2
1 3 4 2
2 1 1 2 2
[ ] [ ]
[ ] [ ]
[ ] [ ][ ]
q A T
M L T A
Fr M LT L
2
2
4
q
F
r
(2) Energy per degree of freedom 1
2 BE k T
(3) Number of particles per unit volume numbern
volume
(4) Charge on particle q = current time [ ] [ ]q AT
Since ( kBT ) is together in every expression , we use 1 2 2
[ ] [ ]Bk T M L T
](https://image.slidesharecdn.com/ude2-160608053437/85/Units-Dimentions-Error-QA-2-3-320.jpg)
![Physics Helpline
L K Satapathy Units Dimensions Error 2
1 2 1 22 3 2 2
1 22 1
1 3 4 2 1 2 2
[ ][ ]
( ) [ ]
[ ]
]
[ ]
[
B
nq L A T
a L L
k T M L T A M L T
False
Correct options = (b , d )
1 2 1 21 3 4 2 1 2 2
1 22 1
2 3 2 2
[ ][ ]
[ ]( ) [ ]
[ ][ ]
Bk T M L T A M L T
b L L
nq L
Tru
A
e
T
1 2 1 22 2 2
3 1 2 3 2
2 3 1 3 4 2 2 1 2 2
[ ]
( ) [ ] [ ]
[ ][ ][
[ ]
]B
q A T
c L L
n k T M L T A L
Fa
M L
lse
T
1 2 1 22 2 2
2 1 2 1
1 3 1 3 4 2 1 1 2 2
[ ]
( ) [ ] [ ]
[ ][ ][ ]
[ ]
B
True
q A T
d L L
n k T M L T A L M L T
Let us examine the given options :](https://image.slidesharecdn.com/ude2-160608053437/85/Units-Dimentions-Error-QA-2-4-320.jpg)
Units Dimentions Error QA 2
- 1. Physics Helpline L K Satapathy Dimensional Consistency Units Dimensions Error 2 [ Mass ] = [ M ] [ Length ] = [ L ] [ Time ] = [ T ] [ Electric Current ] = [ A ] [ Temperature ] = [ K ]
- 2. Physics Helpline L K Satapathy Units Dimensions Error 2 Question : A length scale ( l ) , depends on the permittivity () of a dielectric material , Boltzmann constant ( kB ) , the absolute temperature ( T ) , the number per unit volume (n) of certain charged particles and the charge (q) carried by each of the particles . Which of the following expression(s) for l is(are) dimensionally correct ? 2 2 2 2 2 3 1 3 ( ) ( ) ( ) ( )B B B B k Tnq q q a l b l c l d l k T nq n k T n k T Answer : We will use the following nomenclature for the dimensions of fundamental quantities [ mass ] = [ M ] [ length ] = [ L ] [ time ] = [ T ] [ temperature ] = [ K ] [ electric current ] = [ A ]
- 3. Physics Helpline L K Satapathy Units Dimensions Error 2 1 2 2 1 2 2 1 1 [ ] [ ] [ ] [ ] B energy M L T k M L T K temp K 3 [ ] [ ]n L (1) Electrostatic force between two point charges 2 2 2 1 3 4 2 2 1 1 2 2 [ ] [ ] [ ] [ ] [ ] [ ][ ] q A T M L T A Fr M LT L 2 2 4 q F r (2) Energy per degree of freedom 1 2 BE k T (3) Number of particles per unit volume numbern volume (4) Charge on particle q = current time [ ] [ ]q AT Since ( kBT ) is together in every expression , we use 1 2 2 [ ] [ ]Bk T M L T
- 4. Physics Helpline L K Satapathy Units Dimensions Error 2 1 2 1 22 3 2 2 1 22 1 1 3 4 2 1 2 2 [ ][ ] ( ) [ ] [ ] ] [ ] [ B nq L A T a L L k T M L T A M L T False Correct options = (b , d ) 1 2 1 21 3 4 2 1 2 2 1 22 1 2 3 2 2 [ ][ ] [ ]( ) [ ] [ ][ ] Bk T M L T A M L T b L L nq L Tru A e T 1 2 1 22 2 2 3 1 2 3 2 2 3 1 3 4 2 2 1 2 2 [ ] ( ) [ ] [ ] [ ][ ][ [ ] ]B q A T c L L n k T M L T A L Fa M L lse T 1 2 1 22 2 2 2 1 2 1 1 3 1 3 4 2 1 1 2 2 [ ] ( ) [ ] [ ] [ ][ ][ ] [ ] B True q A T d L L n k T M L T A L M L T Let us examine the given options :
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