To find the values of
m and
n through dimensional analysis, we must ensure that the dimensions on both sides of the proportionality equation are the same. The fundamental dimensions involved here are mass
(M), length
L), and time
(T).
Let's start with the dimensional analysis of each quantity:
The power P has the dimensions of energy per unit time, which is:
[P]=[ML2T−3] The charge q has the dimensions of electric current
(I) times time
(T), giving us:
[q]=[IT]The acceleration a is the change in velocity per unit time, where velocity (v) has dimensions of length per unit time:
[a]=[LT−2]The speed of light
c has the dimensions of velocity, which are:
[c]=[LT−1]Given the proportionality relationship:
P∝The dimensions of the right-hand side are:
[]=[]Simplifying further:
Combining the powers:
[]=[ImLm−nT−m+n] We want this to match the dimensions of power:
[ML2T−3]Therefore, we equate the dimensions of each term on both sides:
For the current
(I) :
m=0 (since there is no current dimension in the power)
For the length
(L) :
m−n=2For the time
(T) :
−m+n=−3Now, we solve these two simultaneous equations:
1.
m−n=22.
−m+n=−3 Adding these two equations:
(m−n)+(−m+n)=2−30=−1So we have:
m−n=2−m+n=−3 Solving these, we get:
begingatheredm=2\n=3endgatheredThus, the correct values are:
Option B
m=2,n=3