082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

368 Chapter 10 Differentiation
Engineeringapplication10.1
Voltageacrossaninductor
The voltage, v, across an inductor with inductance, L, is related to the current, i,
through the inductor by
v =L di
dt
Figure10.11showstherelationshipbetweenmagneticfluxlinespassingthroughthe
coils of aninductor and the current flowing through the inductor.
i
Figure10.11
Schematic diagram ofan
inductorshowing the
relationshipbetween magnetic
fluxlinesand current.
This relationship is a quantification of Faraday’s law which states that the voltage
induced in a coil is proportional to the rate of change of magnetic flux through it. If
the current in a coil is changing then this corresponds to a change in the magnetic
flux through the coil. Note that if di is large then v is large. This is why care has
dt
to be taken when abruptly switching off the current to an inductor because it causes
high voltages tobe generated.
Engineeringapplication10.2
Currentthroughacapacitor
The current,i, through a capacitor with capacitance,C, is related to the voltage, v,
acrossthe capacitorby
i =C dv
dt
Displacement current i D
Conduction current i
i
Figure10.12
Schematic diagram ofacapacitor
showingthe conduction current and
the displacement current.
It may appear confusing to talk of a current flow through a capacitor as no actual
charge flows through the capacitor apart from that caused by any leakage current.
Insteadthereisabuild-upofchargeontheplatesofthecapacitor.Thisinturngives
risetoavoltageacrossthecapacitor.Ifthecurrentflowislargethentherateofchange
ofthisvoltagewillbelarge.Thecurrentflowthroughthecapacitorwiresistermeda