J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’

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                                <s>all such considerations and confine myself to the solution 
of problems on the mutual action of currents, the experimental 
det[erm]ination of coefficients of induction, and the calculation 
of these coefficients from the known form of the circuit.<\s>
In order to bring these results within the range of experimental verification 
I shall next investigate the case of a single current, of two currents and 
of the six currents in the Electric Balance, so as to enable the experimenter 
to determine the values of L, M, N. 

(35) Case of a single circuit

The equation of the current in a circuit whose resistance 
is R and whose coefficient of self=induction is L acted on by 
an external electromotive force [xi] is 

[xi] - Rx = d/dt Lx (13) 

When [xi] is constant, the solution is of the form 

x = b + (a — b) e<sup>- R/L t<\sup> 

where a is the value of the current at the commencement and b is its 
final value. 

The total quantity of electricity which passes in time t where t is great 
is [equation] (14) 

The value of the integral of x<sup>2<\sup>  with respect to the time  
is [equation] (15) 

The actual current changes gradually from the initial value a 
to the final value b, but the values of the integrals of <s>the<\s> 
x and x<sup>2<\sup> are the same as if a steady current of intensity [half](a + b) 
were to flow for a time 2 L/R and were then succeeded by the 
steady current b. The <s>quantity<\s> time 2 L/R is generally so minute 
a fraction of a second that the effects on the galvanometer 
and dynamometer may be calculated as if the impulse were 
instantaneous. 

If the circuit consists of a battery and a coil then when the 
circuit is first completed the effects are the same as if the current 
had only half its final strength during the time 2 L/R. This diminution 

                            
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Manuscript details

Author
James Clerk Maxwell
Reference
PT/72/7
Series
PT
Date
1864
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Cite as

J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’, 1864. From The Royal Society, PT/72/7

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