242
DIMENSIONS OF UNITS.
[626.
founded on the definition of the unit of electricity, Arts. 41, 42, and may be deduced from the equation,
E
=
e
L
2
{\displaystyle {\mathfrak {E}}={\frac {e}{L^{2}}}}
,
which expresses that the resultant force
E
{\displaystyle {\mathfrak {E}}}
e
{\displaystyle e}
L
{\displaystyle L}
e
{\displaystyle e}
L
2
{\displaystyle L^{2}}
[
L
M
e
T
2
]
=
[
e
L
2
]
{\displaystyle \left[{\frac {LM}{eT^{2}}}\right]=\left[{\frac {e}{L^{2}}}\right]}
,
[
m
L
T
]
=
[
M
m
T
]
{\displaystyle \left[{\frac {m}{LT}}\right]=\left[{\frac {M}{mT}}\right]}
,
whence
[
e
]
=
[
L
3
2
M
1
2
T
−
1
]
{\displaystyle [e]=[L^{\frac {3}{2}}M^{\frac {1}{2}}T^{-1}]}
,
m
=
[
L
1
2
M
1
2
]
{\displaystyle m=[L^{\frac {1}{2}}M^{\frac {1}{2}}]}
,
in the electrostatic system.
The electromagnetic system is founded on a precisely similar definition of the unit of strength of a magnetic pole, Art. 374, leading to the equation
H
=
m
L
2
{\displaystyle {\mathfrak {H}}={\frac {m}{L^{2}}}}
.
whence
[
e
L
T
]
=
[
M
e
T
]
{\displaystyle \left[{\frac {e}{LT}}\right]=\left[{\frac {M}{eT}}\right]}
.
[
L
M
m
T
2
]
=
[
m
L
2
]
{\displaystyle \left[{\frac {LM}{mT^{2}}}\right]=\left[{\frac {m}{L^{2}}}\right]}
,
and
[
e
]
=
[
L
1
2
M
1
2
]
{\displaystyle [e]=[L^{\frac {1}{2}}M^{\frac {1}{2}}]}
,
[
m
]
=
[
L
3
2
M
1
2
T
−
1
]
{\displaystyle [m]=[L^{\frac {3}{2}}M^{\frac {1}{2}}T^{-1}]}
,
in the electromagnetic system. From these results we find the dimensions of the other quantities.
626.]
Table of Dimensions.
Dimensions in
Symbol
Electrostatic System
Electromagnetic System
Quantity of electricity
e
{\displaystyle e}
[
L
3
2
M
1
2
T
−
1
]
{\displaystyle [L^{\frac {3}{2}}M^{\frac {1}{2}}T^{-1}]}
[
L
1
2
M
1
2
]
{\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}]}
Line-integral of electromotive force
E
{\displaystyle E}
[
L
1
2
M
1
2
T
−
1
]
{\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}T^{-1}]}
[
L
3
2
M
1
2
T
−
2
]
{\displaystyle [L^{\frac {3}{2}}M^{\frac {1}{2}}T^{-2}]}
Quantity of magnetism
m
{\displaystyle m}
[
L
1
2
M
1
2
]
{\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}]}
[
L
3
2
M
1
2
T
−
1
]
{\displaystyle [L^{\frac {3}{2}}M^{\frac {1}{2}}T^{-1}]}
Electrokinetic momentum of a circuit
p
{\displaystyle p}
Electric current
C
{\displaystyle C}
[
L
3
2
M
1
2
T
−
2
]
{\displaystyle [L^{\frac {3}{2}}M^{\frac {1}{2}}T^{-2}]}
[
L
1
2
M
1
2
T
−
1
]
{\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}T^{-1}]}
Magnetic potential
Ω
{\displaystyle \Omega }
Electric displacement
D
{\displaystyle {\mathfrak {D}}}
[
L
−
1
2
M
1
2
T
−
1
]
{\displaystyle [L^{-{\frac {1}{2}}}M^{\frac {1}{2}}T^{-1}]}
[
L
−
3
2
M
1
2
]
{\displaystyle [L^{-{\frac {3}{2}}}M^{\frac {1}{2}}]}
Surface-density
Electromotive force at a point
E
{\displaystyle {\mathfrak {E}}}
[
L
−
1
2
M
1
2
T
−
1
]
{\displaystyle [L^{-{\frac {1}{2}}}M^{\frac {1}{2}}T^{-1}]}
[
L
1
2
M
1
2
T
−
2
]
{\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}T^{-2}]}
Magnetic induction
B
{\displaystyle {\mathfrak {B}}}
[
L
−
3
2
M
1
2
]
{\displaystyle [L^{-{\frac {3}{2}}}M^{\frac {1}{2}}]}
[
L
−
1
2
M
1
2
T
−
1
]
{\displaystyle [L^{-{\frac {1}{2}}}M^{\frac {1}{2}}T^{-1}]}
Magnetic force
H
{\displaystyle {\mathfrak {H}}}
[
L
1
2
M
1
2
T
−
2
]
{\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}T^{-2}]}
[
L
−
1
2
M
1
2
T
−
2
]
{\displaystyle [L^{-{\frac {1}{2}}}M^{\frac {1}{2}}T^{-2}]}
Strength of current at a point
C
{\displaystyle {\mathfrak {C}}}
[
L
−
1
2
M
1
2
T
−
2
]
{\displaystyle [L^{-{\frac {1}{2}}}M^{\frac {1}{2}}T^{-2}]}
[
L
−
3
2
M
1
2
T
−
1
]
{\displaystyle [L^{-{\frac {3}{2}}}M^{\frac {1}{2}}T^{-1}]}
Vector potential
V
{\displaystyle {\mathfrak {V}}}
[
L
−
1
2
M
1
2
]
{\displaystyle [L^{-{\frac {1}{2}}}M^{\frac {1}{2}}]}
[
L
1
2
M
1
2
T
−
1
]
{\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}T^{-1}]}
![{\displaystyle \left[{\frac {LM}{eT^{2}}}\right]=\left[{\frac {e}{L^{2}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33a120f3298cf0ae97218bb793c9c1ef3a7af46a)
![{\displaystyle \left[{\frac {m}{LT}}\right]=\left[{\frac {M}{mT}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a0bc3cac8c6cdfd57081fae19b1042ef1f5c78)
![{\displaystyle [e]=[L^{\frac {3}{2}}M^{\frac {1}{2}}T^{-1}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c39a5350a901dacc1730187e534130cb74f23188)
![{\displaystyle m=[L^{\frac {1}{2}}M^{\frac {1}{2}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a90946111319549b39e4a6071e47bbc747f40fc)
![{\displaystyle \left[{\frac {e}{LT}}\right]=\left[{\frac {M}{eT}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efc3f198f374498f78ad481a59ea2dc10cf320a5)
![{\displaystyle \left[{\frac {LM}{mT^{2}}}\right]=\left[{\frac {m}{L^{2}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/498ae63cf7ef914c197748e6180f9e911ad81977)
![{\displaystyle [e]=[L^{\frac {1}{2}}M^{\frac {1}{2}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c160aab39fddedd37b857fa8ef734da8aef3e385)
![{\displaystyle [m]=[L^{\frac {3}{2}}M^{\frac {1}{2}}T^{-1}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d0bb0556471272146773dac2d71f6ae23814812)
![{\displaystyle [L^{\frac {3}{2}}M^{\frac {1}{2}}T^{-1}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4decb91453ec9ec22f84c0a3622310bf7690bd7f)
![{\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4b60a9a78c3804bb734eded1de482a208034840)
![{\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}T^{-1}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d61c748743bd91c83fbf727fd4b3d395afad3d5)
![{\displaystyle [L^{\frac {3}{2}}M^{\frac {1}{2}}T^{-2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e53b6f1e40ad959efbccf18a85b940a57428b8ca)
![{\displaystyle [L^{-{\frac {1}{2}}}M^{\frac {1}{2}}T^{-1}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7b86e325d8ee2b319656db20fab549098f966f2)
![{\displaystyle [L^{-{\frac {3}{2}}}M^{\frac {1}{2}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6645850af1a080a26f3d2c479bfcc8d61b349383)
![{\displaystyle [L^{\frac {1}{2}}M^{\frac {1}{2}}T^{-2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/736cf20022ca32479fea8a77c9d19c9f95bf958c)
![{\displaystyle [L^{-{\frac {1}{2}}}M^{\frac {1}{2}}T^{-2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b03cf1f11db80e8f41a650a1c53c1b211262967)
![{\displaystyle [L^{-{\frac {3}{2}}}M^{\frac {1}{2}}T^{-1}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47cf54318480ac1c455cdf971e776b3abd6d5a14)
![{\displaystyle [L^{-{\frac {1}{2}}}M^{\frac {1}{2}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bec625dfacd2ad3ee1f0d2ba27f73cfc2a282806)
