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In the figure a magnetic field $\vec{B}=3t^2\left(-\hat{k}\right)$ is perpendicular to the rectangular loop of sides $1 \,{\rm m}$ and $2\, {\rm m}$ and total resistance $R=3\,\Omega $

(a) Find the magnetic flux through the loop

(b) Find the magnitude of the induced current at $t=2\ {\rm sec}$.

(c) Show the direction of the induced current on the figure.

(a) The magnetic flux through any loop is defined as ${\Phi }_M=\vec{B}.\vec{A}$. so

\[{\Phi }_M=3t^2\left(-\hat{k}\right).2\left(-\hat{k}\right)=6t^2\]

(b) Faraday's law states that the induced emf in any closed loop (circuit) is the negative of the time rate of change of the magnetic flux through it i.e.

\[{\mathcal E}=-\frac{{\rm d}{\Phi }_M}{{\rm d}t}=-12t\]

So the induce current in the loop is $I_{ind}=\left|{\mathcal E}\right|/R\ $so

\[I_{ind}=\frac{12t}{3}=4{\left.t\right|}_{t=2{\rm s}}=8\ {\rm A}\]

Since the magnetic field $B$ is increasing, the magnetic flux through the loop is also increasing. Lenz's law states that the direction of the induced current is such that the induced magnetic field opposes the original flux change. The induced current must be in counter-clockwise.

(c) see part (b)

A circular loop of wire has a $15\,\Omega $ resistance and a radius of $40.0\ {\rm cm}$. There is initially an external magnetic field of $3.0\ $Tesla going __down__ through the loop __into__ the page. At $t=0$, the external magnetic field begins to change in such a way that after $8\ {\rm s}$ it has a magnitude of $1\ {\rm T}$ and is pointing __up__ through the loop __out __of the page). During the interval $t=0$ through $t=8\ {\rm s}$,

(a) Calculate the magnitude of the average induced current through the loop, and

(b) Determine the direction of the average induced current through the loop.

(c) Briefly explain why I had to add the word ``average'' in (a) and (b) above.

A tightly wrapped circular coil of radius $r=0.18\ {\rm m}$ with $100$ turns (wraps) of wire, is sitting on a table. The ends of the wire are attached to a small light bulb whose filament has a resistance of $R=375\ \Omega $. A magnetic field $\vec{B}$, which the entire coil feels, is then turned on. The field makes an angle of $37{}^\circ $ with respect to the normal to the coil and its magnitude changes according to the following graph.

(a) What is the flux $\Phi $ of the magnetic field through a single wrap of the coil at the instant $t=0.100\ {\rm s}$?

(b) What is the time rate of change of the flux through the entire coil from $t=0s$ to $0.1s$?

(c) What is the induced emf during the period from $t=0\, {\rm s}$ to $t=0.1\ {\rm s}$?

(d) What induced current flows through the filament of the light bulb during the interval from $t=0\, {\rm s}\ $to $t=0.1\,{\rm s}$?

(e) Looking at the coil above the table, is the current flow through the coil clockwise or counter-clockwise?

(f) From the time $t=0.1\,{\rm s}$ to $t=0.150\, {\rm s}$, what induced emf is developed across the ends of the filament?

A rectangular loop with width $L$ and a slide wire with mass $m$ are as shown in the figure. A uniform magnetic field $\vec{B}$ is directed perpendicular to the plane of the loop into the plane of the figure. The slide wire is given an initial speed of $v_0$ and then released. There is no friction between the slide wire and the loop, and the resistance of the loop is negligible in comparison to the resistance $R$ of the slide wire.

(a) Obtain an expression for $F$, the magnitude of the force exerted on the wire while it is moving at speed $v$

(b) Show that the distance $x$ that the wire moves before coming to rest is $x=mv_0R/a^2B^2$

$${\mathcal E}=-\frac{d}{dt}{\Phi }_B=-\frac{d}{dt}\left(\vec{B}.\hat{n}dA\right)=-B\frac{dA}{dt} \ \text{where} \ A=Lx $$

\[\Longrightarrow \ {\mathcal E}=-BL\frac{dx}{dt}=-BLv\]

From Ohm's law, we obtain the induced current in the slide wire:

\[i_{induced}=\frac{{\mathcal E}}{R_{wire}}=\frac{BLv}{R}\]

In above the minus sign is omitted, this minus is a sign of Len's law!

When the sliding wire moves to the right, the flux through the loop is increasing so by Lenz's law the direction of induced current must be counter clockwise i.e. $id\vec{l}=iL\hat{j}$. We have a current carrying wire in a uniform magnetic field, so the force exerted on it is,

\[\vec{F}=id\vec{l}\times \vec{B}=iLB\ \left(\hat{j}\times \left(-\hat{k}\right)\right)=-iLB\hat{i}\]

\[\Longrightarrow \vec{F}=\frac{B^2L^2v}{R}\left(-\hat{i}\right)\]

(b) Use the Newton's 2${}^{nd}$ law and definition of the instantaneous acceleration and some manipulation to find instantaneous velocity of the wire as follows:

\[\vec{F}=m\vec{a}=m\frac{dv}{dt}\ \Longrightarrow \ -\frac{B^2L^2v}{R}=m\frac{dv}{dt}\to -\frac{B^2L^2}{Rm}dt=\frac{dv}{v}\]

\[\int^v_{v_0}{\frac{dv}{v}}=-\int{\frac{B^2L^2}{mR}dt}\to \ {\ln \frac{v}{v_0}\ }=-\frac{B^2L^2}{mR}t\ \]

\[\Longrightarrow v=v_0\,{\exp \left(-\frac{B^2L^2}{mR}t\right)\ }\]

We can check its correctness by substituting the initial and final time into it:

$t=0\to v=v_0\ $and $t\to \infty \ \Longrightarrow v=v_0\,e^{-\infty }\to 0$

Now use the definition of the instantaneous velocity to determine the desired distance $x$

\begin{align*}

v=\frac{dx}{dt}\to x &= \int{v\left(t\right)dt}=v_0\int{{\exp \left(-\frac{B^2L^2}{mR}t\right)\ }dt}\\

&=-\frac{mRv_0}{B^2L^2}{\left.{\exp \left(-\frac{B^2L^2}{mR}t\right)\ }\right|}^t_0=\frac{mRv_0}{B^2L^2}\left(1-e^{-\frac{B^2L^2}{mR}t}\right)

\end{align*}

In the limit that $t$ becomes infinitesimally small $t\to \infty $ we get $\ x\left(t\right)=\frac{mR}{B^2L^2}v_0$.

A circular loop of wire is in a region of spatially uniform magnetic field, as shown in the figure below. The magnetic field is directed into the plane of the figure. Determine the direction (clockwise or counterclockwise) of the induced current in the loop when

(a) B is increasing;

(b) B is decreasing;

(c) B is constant with value $B_0$ Explain your reasoning.

A rectangular loop of wire with mass $m$, width $w$, vertical length $l$, and resistance $R$ falls out of a magnetic field under the influence of gravity. The magnetic field is uniform and out of the paper ($\vec{B}=B\hat{x}$) within the area shown (see sketch) and zero outside of that area. At the time shown in the sketch, the loop is exiting the magnetic field at speed $\vec{V}\left(t\right)=V\left(t\right)\hat{z}$ , where ($V\left(t\right)<0$ meaning the loop is moving downward, not upward). Suppose at time $t$ the distance from the top of the loop to the point where the magnetic field goes to zero is $z(t)$ (see sketch)..

(a) What is the relationship between $V(t)$ and $z(t)$? Be careful of your signs here, remember that $z(t)$ is positive and decreasing with time, so $dz(t)/dt<\ 0$.

(b) If we define the area vector $\vec{A}$ to be out of the page, what is the magnetic flux $f_B$ through our circuit at time $t$ (in terms of $z(t)$, not $V(t)$).

(c) What is $d\Phi_B/dt$ ? Is this positive or negative at time $t$? Be careful here, your answer should include $V(t)$(not $z(t)$and remember that $(t)<\ 0$ ).

(d) What is the direction (clockwise or counterclockwise) and magnitude of the induced current on the loop of wire?

(e) What is the direction (into the page or out of the page) of the self-magnetic field due to the induced current inside the circuit loop?

(f) Besides gravity, what other force acts on the loop in the $\pm \ z$-direction? Give the magnitude and direction of this force in terms of the quantities given. (Hint: use $d\vec{F}=Id\vec{L}\times \vec{B}$)

(g) What is the magnitude of the terminal velocity?

Determine the magnitude of the mutual inductance between an infinite straight wire and a circular loop of radius $R$, that has its center a distance $d$ from the axis of the straight wire ($d>R$)

Hint: $\int^{2\pi}_0{\frac{d\theta}{a+b{\cos \theta\ }}}=\frac{2\pi}{\sqrt{a^2-b^2}}$

From Ampere's law $\oint{\vec B.d\vec \ell}=\mu_0I\ $ the magnetic field produced by straight wire at distance $r$ is $B_1=\frac{\mu_0I_1}{2\pi r}$. Using the definition of flux passing through the loop, we have

\[{\Phi }_{21}=\int {\vec{B}}_1.\hat{n}\ da=\frac{\mu_0I_1}{2\pi}\int^R_0{\rho d\rho}\int^{2\pi}_0{\frac{d\theta}{r}}\]

Where $r=d+\rho\,{\cos \theta\ }$ and $da=\rho d\rho \,d\theta$ is the area element in polar coordinates.(see figure). ${\Phi }_{21}$ is the flux produced by the current $I_1$ and passed the loop.

Using the hint $\int^{2\pi}_0{\frac{d\theta}{d+\rho\,{\cos \theta\ }}=2\pi/\sqrt{d^2-\rho^2}}$ ,one can show that ${\Phi }_{21}=\mu_0I_1\ \int^R_0{\frac{\rho d\rho}{\sqrt{d^2-\rho^2}}}\ \ $

For determining the integral above we use the change of variables

let $u\equiv d^2-\rho^2$ , $\rho d\rho=-\frac{du}{2}$

\[ \Longrightarrow {\Phi }_{21}=\mu_0I_1\ \int^R_0{\frac{\rho d\rho}{\sqrt{d^2-\rho^2}}}=-\frac{1}{2}\mu_0I_1 \int^{d^{2}-R^{2}}_{d^{2}}{\frac{du}{\sqrt{u}}}\ =\mu_0I_1 \left(d-\sqrt{d^2-R^2}\right)\]

so the mutual inductance, by definition, is

\[M_{21}=\frac{{\Phi }_{21}}{I_1}=\mu_0\left(d-\sqrt{d^2-R^2}\right)\]

A slide wire generator is shown in the figure where there is a perpendicular constant magnetic field $B$, uniform everywhere. Assume that the total resistance of the rectangular loop to be only due to the sliding part given by $R$. Now, if the sliding rod (length $L$) is pulled at a constant velocity $v$ shown in the figure, find:

(a) The rate at which energy is dissipated around this loop.

(b) The rate at which mechanical work is done to move the rod through the magnetic field. The dimensions of the stationary parts of the loop are not specified.

A loop of wire with $N=100$ turns and a radius of $3\ {\rm m}$ is situated in a decreasing magnetic field of $1\ {\rm T/sec}$ whose direction is depicted below.

(a) What is the direction of the induced current? Draw the direction of the diagram and explain your reasoning.

(b) What is the magnitude of the current if the resistance of the loop is $10\ \Omega $?

What is the emf induced in a solenoid with an inductance of ${\rm 0.25\ H}$ if the current is uniformly reduced from ${\rm 2.0\ A}$ to ${\rm 0\ A}$ in ${\rm 1/16}$ of a second?

A conducting bar of length $D$ rotates with angular frequency $\omega$ about a pivot $P$ at one end of the bar (see the figure at right). The other end of the bar is in slipping contact with a stationary conducting wire in the shape of a circle (we only show part of that circle to keep the drawing simple). Between point $P$ and the circular wire there is a resistor $R$ as shown. Thus the bar, the resistor, and the wire form a closed conducting loop. The resistance of the bar and the circular wire are negligibly small.

There is a uniform magnetic field $B$ perpendicular to the plane of the conducting wire, as shown. What is the induced current in the loop? Express your answer in terms of $D,\ \omega,\ R,\ $and $B$.

A ``rail gun projectile launcher'' is shown at right. A large current moves in a closed loop composed of fixed rails, a power supply, and a very light, almost frictionless bar touching the rails. A magnetic field is perpendicular to the plane of the circuit.

(a) If the bar has a length ${\rm d=24\ cm}$, a mass of ${\rm 1.5\ g}$, and is placed in a field of ${\rm 1.8\ T}$, what constant current flow is needed to accelerate the bar from rest to a speed of ${\rm 25\ m/}{\rm s}$ in a distance of ${\rm 1.0\ m}$?

(b) In what direction must the field point?

Category : Electromagnetic Induction

**Most useful formula in Waves:**

Faraday's law:

\[\mathcal E=-\frac{d\Phi_B}{dt}\]

induced electric field:

\[\oint \vec E \cdot d\vec \ell=-\frac{d\Phi_B}{dt}\]

$emf$ produced in a conductor with length $L$ moves in uniform $B$ with velocity $v$

\[\mathcal E=vBL\]

Magnetic flux through surface $A$:

\[\Phi_B=\vec B \cdot \vec A\]

Definition of self inductance:

\[L=\frac{\Phi_B}{I}\]

Self-inductance of a solenoid of length $\ell$, cross sectional area $A$ and number of turns per unit length $n$:

\[L=\mu_0 n^2 A \ell\]

Energy stored in an inductor:

\[U=\frac{1}{2}LI^2\]

Number Of Questions : 12

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