Moving Charges And Magnetism

Chapter 4

MOVING CHARGES AND MAGNETISM

MCQ I

4.1 Two charged particles traverse identical helical paths in a completely opposite sense in a uniform magnetic field $\mathbf{B}=B_0 \hat{\mathbf{k}}$.

(a) They have equal $z$-components of momenta.

(b) They must have equal charges.

(d) The charge to mass ratio satisfy : $(\frac{e}{m})_1+(\frac{e}{m})_2 = 0$.

4.2 Biot-Savart law indicates that the moving electrons (velocity v) produce a magnetic field $\mathbf{B}$ such that

(a) $\mathbf{B} \perp \mathbf{v}$.

(b) $\mathbf{B} | \mathbf{v}$.

(d) it is along the line joining the electron and point of observation.

4.3 A current carrying circular loop of radius $R$ is placed in the $x$ - $y$ plane with centre at the origin. Half of the loop with $x>0$ is now bent so that it now lies in the $y-z$ plane.

(a) The magnitude of magnetic moment now diminishes.

(b) The magnetic moment does not change.

(d) The magnitude of $\mathbf{B}$ at $(0.0 . z), z»R$ is unchanged.

4.4 An electron is projected with uniform velocity along the axis of a current carrying long solenoid. Which of the following is true?

(a) The electron will be accelerated along the axis.

(b) The electron path will be circular about the axis.

(d) The electron will continue to move with uniform velocity along the axis of the solenoid.

4.5 In a cyclotron, a charged particle

(a) undergoes acceleration all the time.

(b) speeds up between the dees because of the magnetic field.

(d) slows down within a dee and speeds up between dees.

4.6 A circular current loop of magnetic moment $M$ is in an arbitrary orientation in an external magnetic field $\mathbf{B}$. The work done to rotate the loop by $30^{\circ}$ about an axis perpendicular to its plane is

(a) $M B$.

(b) $\sqrt{3} \frac{M B}{2}$.

(d) zero.

MCQ II

4.7 The gyro-magnetic ratio of an electron in an $H$-atom, according to Bohr model, is

(a) independent of which orbit it is in.

(b) negative.

(d) increases with the quantum number $n$.

4.8 Consider a wire carrying a steady current, I placed in a uniform magnetic field $\mathbf{B}$ perpendicular to its length. Consider the charges inside the wire. It is known that magnetic forces do no work. This implies that,

(a) motion of charges inside the conductor is unaffected by $\mathbf{B}$ since they do not absorb energy.

(b) some charges inside the wire move to the surface as a result of $\mathbf{B}$.

(d) if the wire moves under the influence of $\mathbf{B}$, no work is done by the magnetic force on the ions, assumed fixed within the wire.

4.9 Two identical current carrying coaxial loops, carry current $I$ in an opposite sense. A simple amperian loop passes through both of them once. Calling the loop as $C$,

(a) $\oint_c \mathbf{B} \cdot \mathbf{d l}=\mp 2 \mu_0 I$.

(b) the value of $\oint_c \mathbf{B} . \mathbf{d l}$ is independent of sense of C.

(d) B vanishes everywhere on C.

4.10 A cubical region of space is filled with some uniform electric and magnetic fields. An electron enters the cube across one of its faces with velocity $\mathbf{v}$ and a positron enters via opposite face with velocity -v. At this instant,

(a) the electric forces on both the particles cause identical accelerations.

(b) the magnetic forces on both the particles cause equal accelerations.

(d) the motion of the centre of mass (CM) is determined by $\mathbf{B}$ alone.

4.11 A charged particle would continue to move with a constant velocity in a region wherein,

(a) $\mathbf{E}=0, \mathbf{B} \neq 0$.

(b) $\mathbf{E} \neq 0, \mathbf{B} \neq 0$.

(d) $\mathbf{E}=0, \mathbf{B}=0$.

VSA

4.12 Verify that the cyclotron frequency $\omega=e B / m$ has the correct dimensions of $[T]^{-1}$.

4.13 Show that a force that does no work must be a velocity dependent force.

4.14 The magnetic force depends on $\mathbf{v}$ which depends on the inertial frame of reference. Does then the magnetic force differ from inertial frame to frame? Is it reasonable that the net acceleration has a different value in different frames of reference?

4.15 Describe the motion of a charged particle in a cyclotron if the frequency of the radio frequency ( $rf$ ) field were doubled.

4.16 Two long wires carrying current $I_1$ and $I_2$ are arranged as shown in Fig. 4.1. The one carrying current $I_1$ is along is the $x$-axis. The other carrying current $I_2$ is along a line parallel to the $y$-axis given by $x=0$ and $z=d$. Find the force exerted at $O_2$, because of the wire along the $x$-axis.

alt text

SA

4.17 A current carrying loop consists of 3 identical quarter circles of radius $R$, lying in the positive quadrants of the $x-y, y-z$ and $z-x$ planes with their centres at the origin, joined together. Find the direction and magnitude of $\mathbf{B}$ at the origin.

4.18 A charged particle of charge $e$ and mass $m$ is moving in an electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$. Construct dimensionless quantities and quantities of dimension $[T]^{-1}$.

4.19 An electron enters with a velocity $\mathbf{v}=v_0 \mathbf{i}$ into a cubical region (faces parallel to coordinate planes) in which there are uniform electric and magnetic fields. The orbit of the electron is found to spiral down inside the cube in plane parallel to the $x-y$ plane. Suggest a configuration of fields $\mathbf{E}$ and $\mathbf{B}$ that can lead to it.

4.20 Do magnetic forces obey Newton’s third law. Verify for two current elements $\boldsymbol{{}d} \boldsymbol{{}l}_1=d \boldsymbol{{}l} \hat{\mathbf{i}}$ located at the origin and $\boldsymbol{{}d} \boldsymbol{{}l}_2=d l \hat{\mathbf{j}}$ located at $(0, R, 0)$. Both carry current $I$.

4.21 A multirange voltmeter can be constructed by using a galvanometer circuit as shown in Fig. 4.2. We want to construct a voltmeter that can measure 2V, 20V and 200V using a galvanometer of resistance $10 \varOmega $ and that produces maximum deflection for current of 1 mA. Find $ R_1, R_2 $ and $ R_3 $ that have to be used. Fig. 4.2

4.22 A long straight wire carrying current of 25A rests on a table as shown in Fig. 4.3. Another wire PQ of length 1m, mass $2.5 g$ carries the same current but in the opposite direction. The wire $PQ$ is free to slide up and down. To what height will PQ rise? Fig. 4.3

LA

4.23 A 100 turn rectangular coil $ABCD$ (in $XY$ plane) is hung from one arm of a balance (Fig. 4.4). A mass $500 g$ is added to the other arm to balance the weight of the coil. A current 4.9 A passes through the coil and a constant magnetic field of $0.2 T$ acting inward (in $x z$ plane) is switched on such that only arm $CD$ of length $1 cm$ lies in the field. How much additional mass ’ $m$ ’ must be added to regain the balance? Fig. 4.4

4.24 A rectangular conducting loop consists of two wires on two opposite sides of length $l$ joined together by rods of length $d$. The wires are each of the same material but with cross-sections differing by a factor of 2. The thicker wire has a resistance $R$ and the rods are of low resistance, which in turn are connected to a constant voltage source $V_0$. The loop is placed in uniform a magnetic field $\mathbf{B}$ at $45^{\circ}$ to its plane. Find $\tau$, the torque exerted by the magnetic field on the loop about an axis through the centres of rods.

4.25 An electron and a positron are released from $(0,0,0)$ and $(0,0,1.5 R)$ respectively, in a uniform magnetic field $\mathbf{B}=B_0 \hat{\mathbf{i}}$, each with an equal momentum of magnitude $p=e B R$. Under what conditions on the direction of momentum will the orbits be nonintersecting circles?

4.26 A uniform conducting wire of length $12 a$ and resistance $R$ is wound up as a current carrying coil in the shape of (i) an equilateral triangle of side $a$; (ii) a square of sides $a$ and, (iii) a regular hexagon of sides $a$. The coil is connected to a voltage source $V_0$. Find the magnetic moment of the coils in each case.

4.27 Consider a circular current-carrying loop of radius $R$ in the $x-y$ plane with centre at origin. Consider the line intergral $\mathfrak{I}(L)=|\int _{-L}^{L} \mathbf{B} \cdot \mathbf{d l}|$ taken along $z$-axis.

(a) Show that $\mathfrak{I}(L)$ monotonically increases with $L$.

(b) Use an appropriate Amperian loop to show that $\mathfrak{I}(\infty)=\mu_0 I$, where $I$ is the current in the wire.

(d) Suppose we replace the circular coil by a square coil of sides $R$ carrying the same current $I$. What can you say about $\mathfrak{I}(L)$ and $\mathfrak{I}(\infty)$ ?

4.28 A multirange current meter can be constructed by using a galvanometer circuit as shown in Fig. 4.5. We want a current meter that can measure $10 mA, 100 mA$ and $1 A$ using a galvanometer of resistance $10 \Omega$ and that prduces maximum deflection for current of $1 mA$. Find $S_1, S_2$ and $S_3$ that have to be used

Fig. 4.5

4.29 Five long wires A, B, C, D and E, each carrying current $I$ are arranged to form edges of a pentagonal prism as shown in Fig. 4.6. Each carries current out of the plane of paper.

(a) What will be magnetic induction at a point on the axis $O$ ? Axis $ _{E}$ is at a distance $R$ from each wire.

(b) What will be the field if current in one of the wires (say A) is switched off?

alt text

Chapter 4

4.1 (d)

4.2 (a)

4.3 (a)

4.4 (d)

4.5 (a)

4.6 (d)

4.7 (a), (b)

4.8 (b), (d)

4.9 (b), (c)

4.10 (b), (c), (d)

4.11 (a), (b), (d)

4.12 For a charge particle moving perpendicular to the magnetic field:

$\frac{m v^{2}}{R}=q v B$

$\therefore \frac{q B}{m}=\frac{v}{R}=\omega$

$\therefore[\omega]=[\frac{q B}{m}]=[\frac{v}{R}]=[T]^{-1}$.

4.13 $ dW=\mathbf{F} . d \mathbf{1}=0$

$\Rightarrow \mathbf{F} . \mathbf{v} d t=0$

$\Rightarrow \mathbf{F} . \mathbf{v}=0$

$\mathbf{F}$ must be velocity dependent which implies that angle between $\mathbf{F}$ and $\mathbf{v}$ is $90^{\circ}$. If $\mathbf{v}$ changes (direction) then (directions) $\mathbf{F}$ should also change so that above condition is satisfied.

4.14 Magnetic force is frame dependent. Net acceleraton arising from this is however frame independent (non - relativistic physics)for inertial frames.

4.15 Particle will accelerate and decelerate altenatively. So the radius of path in the Dee’s will remain unchanged.

4.16 At $O_2$, the magnetic field due to $I_1$ is along the $y$-axis. The second wire is along the $y$-axis and hence the force is zero.

4.17

$\mathbf{B}=\frac{1}{4}(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}) \frac{\mu_0 I}{2 R}$

4.18 No dimensionless quantity $[T]^{-1}=[\omega]=[\frac{e B}{m}]$

4.19 $\mathbf{E}=E_0 \hat{\mathbf{i}}, E_O>0, \mathbf{B}=B_0 \hat{\mathbf{k}}$

4.20 Force due to $d \mathbf{1} _{\mathbf{2}}$ on $d \mathbf{1} _{\mathbf{1}}$ is zero.

Force due to $d \mathbf{1} _{\mathbf{1}}$ on $d \mathbf{1} _{\mathbf{2}}$ is non-zero.

4.21 $ i_G(G+R_1)=2$ for $2 V$ range

$ \begin{aligned} & i_G(G+R_1+R_2)=20 \text{ for } 20 V \text{ range } \\ & \text{ and } i_G(G+R_1+R_2+R_3)=200 \text{ for } 200 V \text{ range } \\ & \text{ Gives } \quad R_1=1990 \Omega \\ & \qquad \begin{matrix} R_2=18 k \Omega \\ \text{ and } \quad R_3=180 k \Omega \end{matrix} \end{aligned} $

4.22 $F=B I l$ Sin $\theta=B I l$

$B=\frac{\mu_0 I}{2 \pi h}$

$F=m g=\frac{\mu_0 I^{2} l}{2 \pi h}$

$ \begin{aligned} h= & \frac{\mu_o I^{2} l}{2 \pi m g}=\frac{4 \pi \times 10^{-7} \times 250 \times 25 \times 1}{2 \pi \times 2.5 \times 10^{-3} \times 9.8} \\ & =51 \times 10^{-4} \\ & h=0.51 cm \end{aligned} $

4.23 When the field is off $\sum \tau=0$

$M g l=W _{\text{coil }} l$

$500 g l=W _{\text{coil }} l$

$W _{\text{coil }}=500 \times 9.8 N$

When the magnetic field is switched on

$M g l+m g l=W _{\text{coil }} l+I B L \sin 90^{\circ} l$

$m g l=B I L l$

$m=\frac{B I L}{g}=\frac{0.2 \times 4.9 \times 1 \times 10^{-2}}{9.8}=10^{-3} kg$

$=1 g$

4.24
$F_1=i_1 l B=\frac{V_O}{R} l B \quad \tau_1=\frac{d}{2 \sqrt{2}} F_1=\frac{V_0 l d B}{2 \sqrt{2} R}$

$F_2=i_2 l B=\frac{V_0}{2 R} l B$

$ \tau_2=\frac{d}{2 \sqrt{2}} F_2=\frac{V_0 l d B}{4 \sqrt{2} R} $

Net torque $\tau=\tau_1-\tau_2$

$\tau=\frac{1}{4 \sqrt{2}} \frac{V_0 A B}{R}$

alt text

4.25 As $\mathbf{B}$ is along the $x$ axis, for a circular orbit the momenta of the two particles are in the $y-z$ plane. Let $\mathbf{p} _1$ and $\mathbf{p} _2$ be the momentum of the electron and positron, respectively. Both of them define a circle of radius $R$. They shall define circles of opposite sense. Let $\mathbf{p} _1$ make an angle $\theta$ with the $y$ axis $\mathbf{p} _2$ must make the same angle. The centres of the repective circles must be perpendicular to the momenta and at a distance $R$. Let the center of the electron be at $Ce$ and of the positron at $C p$. The coordinates of $Ce$ is

The coordinates of $Ce$ is $C e \equiv(0,-R \sin \theta, R \cos \theta)$

The coordinates of $Cp$ is

alt text

$C p \equiv(0,-R \sin \theta, \frac{3}{2} R-R \cos \theta)$

The circles of the two shall not overlap if the distance between the two centers are greater than $2 R$.

Let $d$ be the distance between $Cp$ and $Ce$.

Then $d^{2}=(2 RSin \theta)^{2}+(\frac{3}{2} R-2 R \cos \theta)^{2}$

$=4 R^{2} Sin^{2} \theta+\frac{9}4^{2} R-6 R^{2} \cos \theta+4 R^{2} \cos ^{2} \theta$

$=4 R^{2}+\frac{9}{4} R^{2}-6 R^{2} \cos \theta$

Since $d$ has to be greater than $2 R$

$d^{2}>4 R^{2}$

$\Rightarrow 4 R^{2}+\frac{9}{4} R^{2}-6 R^{2} \cos \theta>4 R^{2}$

$\Rightarrow \frac{9}{4}>6 \cos \theta$

Or, $\cos \theta<\frac{3}{8}$.

2.26

Area: $A=\frac{\sqrt{3}}{4} a^{2}$

$ A=a^{2} \quad A=\frac{3 \sqrt{3}}{4} a^{2} $

Current $I$ is same for all

Magnetic moment $m=n$ IA

$ \therefore m=I a^{2} \sqrt{3} \quad 3 a^{2} I \quad 3 \sqrt{3} a^{2} I $

(Note: $m$ is in a geometric series)

4.27 (a) $B(z)$ points in the same direction on $z$ - axis and hence $J(L)$ is a monotonically increasing function of $L$.

(b) $J(L)+$ Contribution from large distance on contour $C=\mu_0 I$

$\therefore$ asL $arrow \infty$

Contribution from large distance $arrow O(asB \sim 1 / r^{3})$

$J(\infty)-\mu_0 I$

$\int _{-\infty}^{\infty} B_z d z=\int _{-\infty}^{\infty} \frac{\mu_0 I R^{2}}{2(z^{2}+R^{2})^{3 / 2}} d z$

Put $z=R \tan \theta \quad d z=R \sec ^{2} \theta d \theta$

$\therefore \int _{-\infty}^{\infty} B_z d z=\frac{\mu_0 I}{2} \int _{-\pi / 2}^{\pi / 2} \cos \theta d \theta=\mu_0 I$

(d) $B(z) _{\text{square }}<B(z) _{\text{circular coil }}$

$\therefore \mathfrak{I}(L) _{\text{square }}<\mathfrak{I}(L) _{\text{circular coil }}$

But by using arguments as in (b)

$\mathfrak{I}(\infty) _{\text{square }}=\mathfrak{I}(\infty) _{\text{circular }}$

4.28

$i_G \cdot G=(i_1-i_G)(S_1+S_2+S_3)$

for $i_1=10 mA$

$i_G(G+S_1)=(i_2-i_G)(S_2+S_3)$

for $i_2=100 mA$

and $i_G(G+S_1+S_2)=(i_3-i_G)(S_3) \quad$ for $i_3=1 A$

gives $S_1=1 W, S_2=0.1 W$ and $S_3=0.01 W$

4.29 (a) zero

(b) $\frac{\mu_0}{2 \pi} \frac{i}{R}$ perpendicular to $AO$ towards left.