Electromagnetic Induction Concept Page - 4

Example

Circuit rotating perpendicular to the plane of magnetic field

A coil has 1000$1000$ turns and 500$500$ cm2$c{m}^2$ as it's area. The plane of the coil is placed at right angles to a magnetic induction field of 2×10−5Wb/m2$2\times {10}^{-5}Wb/m^2$. The coil is rotated through 1800${180}^0$ in 0.2$0.2$ seconds. The average emf induced in the coil (in milli volts) is:The induced emf will be given by
emf=△ϕ△t$emf={\displaystyle \frac{\mathrm{△}\varphi }{\mathrm{△}t}}$
          =2ϕt$={\displaystyle \frac{2\varphi }{t}}$
          =2×B×A×nt$={\displaystyle \frac{2\times B\times A\times n}{t}}$
          =2×2×10−5×500×10−4×10000.2$={\displaystyle \frac{2\times 2\times {10}^{-5}\times 500\times {10}^{-4}\times 1000}{0.2}}$
          =10×10−3V=10$=10\times {10}^{-3}V=10$ mV

Example

Moving conductor with a constant velocity in a circuit with capacitor

In the shown figure, there are two long fixed parallel conducting rails (having negligible resistance) and are separated by distance L. A uniform rod(cd) of resistance R and mass M is placed at rest on frictionless rails. Now at time t=0,$t=0,$ a capacitor having charge Q0$Q_0$ and capacitance C is connected across rails at ends a and b such that current in rod(cd) is from c towards d and the rod is released. A uniform and constant magnetic field having magnitude B exists normal to plane of paper as shown. (Neglect acceleration due to gravity)At any instant t,$t,$ the charge on capacitor q, velocity of rod v$v$ and the current I$I$ through rod are as shown

∴Mdvdt=BIL=B(−dqdt)L....(1)$\therefore M{\displaystyle \frac{dv}{dt}}=BIL=B(-{\displaystyle \frac{dq}{dt}})L....(1)$
⇒∫v0Mdv=−∫qQ0BLdq$\Rightarrow {\int }_0^{v}Mdv=-{\int }_{Q_0}^{q}BLdq$
solving we get q, q=Q0−MvBL................(2)$q,q=Q_0-{\displaystyle \frac{Mv}{BL}}................(2)$
AlsoqC=BLV+IR=BLv−Rdqdt..........(3)${\displaystyle \frac{q}{C}}=BLV+IR=BLv-R{\displaystyle \frac{dq}{dt}}..........(3)$
From equations (1)$(1)$ and (3)$(3)$
qC=BLv+MRBLdvdt.................(4)${\displaystyle \frac{q}{C}}=BLv+{\displaystyle \frac{MR}{BL}}{\displaystyle \frac{dv}{dt}}.................(4)$
from (4) when dvdt=0⇒qC=Blv....(5)${\displaystyle \frac{dv}{dt}}=0\Rightarrow {\displaystyle \frac{q}{C}}=Blv....(5)$
From (2) and (5). At instant acceleration is zero
v=Q0LBM+B2L2C$v={\displaystyle \frac{Q_{0}LB}{M+B^2{L}^{2}C}}$ and q=B2L2CQ0M+B2L2C$q={\displaystyle \frac{B^2{L}^{2}C{Q}_0}{M+B^2{L}^{2}C}}$

Example

Moving conductor in a circuit using Newton's Laws

A magnetic field B⃗ =Boyak^$\overrightarrow{B}=B_o{\displaystyle \frac{y}{a}}\hat{k}$ is out of the x-y plane, were Bo$B_o$ and a$a$ are positive constants. A square loop PQRS of side a$a$, mass m$m$ and resistance R$R$ in x-y plane starts falling under gravity. The terminal velocity of the loop then must be (Given acceleration due to gravity =g$=g$):
When the side SR of the loop is at a distance y$y$ from x-axis, let v$v$ be its velocity along y-axis induced emf across RS=Boya⋅av=Boyv${\displaystyle \frac{B_{o}y}{a}}\cdot av=B_{o}yv$ with S at higher potential.
induced emf across PQ = Bo(y+a)a⋅av=Bo(y+a)v$B_o{\displaystyle \frac{(y+a)}{a}}\cdot av=B_o(y+a)v$,
with Q$Q$ at higher potential
∴$\therefore$ net emf in the loop = Bov(y+a−y)=Boav$B_{o}v(y+a-y)=B_{o}av$
The induced current i = BoavR$\frac{B_{o}av}{R}$, R - resistance of the loop.
The forces due to magnetic field on PS and QR cancel each other. Hence magnetic force on loop
FM→=Bo(y+a−y)a⋅Boa2vR=B2oa2Rvj^$\overrightarrow{F_M}=B_o{\displaystyle \frac{(y+a-y)}{a}}\cdot {\displaystyle \frac{B_o{a}^{2}v}{R}}={\displaystyle \frac{B_o^2{a}^2}{R}}v\hat{j}$
∴$\therefore$ net force on loop = (B2oa2Rv−mg)↑$({\displaystyle \frac{B_o^2{a}^2}{R}}v-mg)\uparrow$
∴mdvdt=mg−B2oa2Rv$\therefore m{\displaystyle \frac{dv}{dt}}=mg-{\displaystyle \frac{B_o^2{a}^2}{R}}v$
Terminal velocity vo$v_o$ is that at which net force = 0
∴vo=mgRB2oa2$\therefore v_o={\displaystyle \frac{mgR}{B_o^2{a}^2}}$

Definition

Inductance

Inductance is the property of an electrical conductor by which a change in current through it induces an electromotive force in both the conductor itself. It is the constant of proportionality that relates flux linkage with current.

Definition

Units of Resistance,Capacitance and Inductance

SI unit of resistance is Ω$\mathrm{\Omega }$ (Ohm)
R=ρlA$R={\displaystyle \frac{\rho l}{A}}$

SI unit of capacitance is F$F$ (farad).
A 1 farad capacitor, when charged with 1 coulomb of electrical charge, has a potential difference of 1 volt between its plates.
SI unit of Inductance is H$H$(henry).
One henry is the equivalent kgm2s−2A−2$kg{m}^2{s}^{-2}{A}^{-2}$

Definition

Dependency of geometric parameters on inductance of a coil

NUMBER OF WIRE WRAPS, OR TURNS IN THE COIL: All other factors being equal, a greater number of turns of wire in the coil results in greater inductance; fewer turns of wire in the coil results in less inductance.COIL AREA: All other factors being equal, greater coil area (as measured looking lengthwise through the coil, at the cross-section of the core) results in greater inductance; less coil area results in less inductance.COIL LENGTH: All other factors being equal, the longer the coils length, the less inductance; the shorter the coils length, the greater the inductance.

Definition

Inductor

An inductor is also called a coil or a reactor, is a passive two-terminal electrical component that stores electrical energy in a magnetic field when electric current is flowing through it. An inductor typically consists of an electric conductor, such as a wire, that is wound into a coil.

Definition

Mutual inductance

If two coils of wire are brought into close proximity with each other so the magnetic field from one links with the other, a voltage will be generated in the second coil as a result. This is called mutual inductance when voltage impressed upon one coil induces a voltage in another.

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