3. If energy, E=G'h".", where G is the universal gravitational constant, h is Planck's constant and c is
the velocity of light, then find the value of l, m and n using dimensions.
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Explanation:
Dimension of Energy, E=[ML
2
T
−2
]
Dimension of gravitational constant, G=[M
−1
L
3
T
−2
]
Dimension of Planck's constant, h=[ML
2
T
−1
]
Dimension of speed of light, c=[M
0
LT
−1
]
Given :
E=G
p
h
q
c
r
∴ [ML
2
T
−2
]=[M
−1
L
3
T
−2
]
p
× [ML
2
T
−1
]
q
×[M
0
LT
−1
]
r
⟹ [ML
2
T
−2
] =[M
(−p+q)
L
(3p+2q+r)
T
(−2p−q−r)
]
Equating both sides, we get:
−p+q=1 .............(1)
3p+2q+r=2 .............(2)
−2p−q−r=−2 .............(3)
Adding (2) and (3), we get: p+q=0 .......(4)
Solving (1) and (4),
⟹p=−
2
1
and q=
2
1
Now from (2),
3×(−
2
1
)+2×
2
1
+r=2
⟹r=
2
5
How s
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