(B) Using the work-energy theorem: $W_{net} = \Delta K$$qE(2a) = \frac{1}{2}m(2v)^2 - \frac{1}{2}mv^2 = \frac{3}{2}mv^2$$E = \frac{3mv^2}{4qa}$. Thus,

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(B) Using the work-energy theorem: $W_{net} = \Delta K$$qE(2a) = \frac{1}{2}m(2v)^2 - \frac{1}{2}mv^2 = \frac{3}{2}mv^2$$E = \frac{3mv^2}{4qa}$. Thus,$(A)$ is correct.Rate of work done by electric field $P_E = \vec{F}_E \cdot \vec{v} = (qE\hat{i}) \cdot (v\hat{i}) = qEv = q\left(\frac{3mv^2}{4qa}\right)v = \frac{3mv^3}{4a}$. Thus,$(B)$ is correct.At $Q$,velocity is $-2v\hat{j}$. Electric force is $qE\hat{i}$. Since $\vec{F}_E \perp \vec{v}$,power by electric field is $0$. Magnetic force is always perpendicular to velocity,so power by magnetic field is $0$. Thus,$(C)$ is correct.Angular momentum $L = \vec{r} \times \vec{p}$. At $P$,$\vec{r}_P = a\hat{j}$,$\vec{p}_P = mv\hat{i}$,so $L_P = (a\hat{j}) \times (mv\hat{i}) = -mav\hat{k}$. Magnitude $|L_P| = mav$.At $Q$,$\vec{r}_Q = 2a\hat{i}$,$\vec{p}_Q = -2mv\hat{j}$,so $L_Q = (2a\hat{i}) \times (-2mv\hat{j}) = -4mav\hat{k}$. Magnitude $|L_Q| = 4mav$.Difference $= 4mav - mav = 3mav$. Thus,$(D)$ is incorrect.

Class 12 Physics Moving Charges and Magnetism Mix Examples-Moving Charges and Magnetism

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$A$ charged particle of mass $m$ and charge $q$ moving under the influence of a uniform electric field $E\hat{i}$ and a uniform magnetic field $B\hat{k}$ follows a trajectory from point $P$ to $Q$ as shown in the figure. The velocities at $P$ and $Q$ are respectively $v\hat{i}$ and $-2v\hat{j}$. Which of the following statements $(A, B, C, D)$ are correct? (Trajectory shown is schematic and not to scale)
$(A)$ $E = \frac{3}{4}\left(\frac{mv^{2}}{qa}\right)$
$(B)$ Rate of work done by the electric field at $P$ is $\frac{3}{4}\left(\frac{mv^{3}}{a}\right)$
$(C)$ Rate of work done by both the fields at $Q$ is zero
$(D)$ The difference between the magnitude of angular momentum of the particle at $P$ and $Q$ is $2mav$.

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A
$(A), (B), (C), (D)$
B
$(A), (B), (C)$
C
$(B), (C), (D)$
D
$(A), (C), (D)$

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Moving Charges and Magnetism

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Mix Examples-Moving Charges and Magnetism

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DifficultAIEEE 2012

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As the temperature of the hot junction increases, the thermo $e.m.f.$

Medium

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Six point charges,each of magnitude $q$,are arranged in different manners as shown in the image. In each case,a point $M$ and a line $PQ$ passing through $M$ are shown. Let $E$ be the electric field and $V$ be the electric potential at $M$ (potential at infinity is zero) due to the given charge distribution when it is at rest. Now,the whole system is set into rotation with a constant angular velocity about the line $PQ$. Let $B$ be the magnetic field at $M$ and $\mu$ be the magnetic moment of the system in this condition. Assume each rotating charge to be equivalent to a steady current. Match the conditions in Column $I$ with the configurations in Column $II$.
Column $I$ Column $II$
$(A)$ $E=0$ $(p)$ Charges at corners of a regular hexagon. $M$ is the centre. $PQ$ is perpendicular to the plane.
$(B)$ $V \neq 0$ $(q)$ Charges on a line perpendicular to $PQ$ at equal intervals. $M$ is the mid-point.
$(C)$ $B=0$ $(r)$ Charges on two coplanar concentric rings. $M$ is the common centre. $PQ$ is perpendicular to the plane.
$(D)$ $\mu \neq 0$ $(s)$ Charges at corners and mid-points of a rectangle. $M$ is the centre. $PQ$ is parallel to the longer sides.
$(t)$ Charges on two coplanar,identical rings. $M$ is the mid-point between centres. $PQ$ is perpendicular to the line joining centres.

DifficultIIT 2009

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One of the two small circular coils (neither having any self-inductance) is suspended with a $V$-shaped copper wire,with its plane horizontal. The other coil is placed just below the first one with its plane horizontal. Both coils are connected in series with a $dc$ supply. The coils are found to attract each other with a force. Which one of the following statements is incorrect?

DifficultJEE MAIN 2013

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Two parallel beams of protons and electrons,carrying equal currents,are fixed at a separation $d$. The protons and electrons move in opposite directions. $P$ is a point on a line joining the beams,at a distance $x$ from the proton beam. The magnetic field at $P$ is $B$. If $B$ is plotted against $x$,which of the following best represents the resulting curve?

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Column $I$	Column $II$
$(A)$ $E=0$	$(p)$ Charges at corners of a regular hexagon. $M$ is the centre. $PQ$ is perpendicular to the plane.
$(B)$ $V \neq 0$	$(q)$ Charges on a line perpendicular to $PQ$ at equal intervals. $M$ is the mid-point.
$(C)$ $B=0$	$(r)$ Charges on two coplanar concentric rings. $M$ is the common centre. $PQ$ is perpendicular to the plane.
$(D)$ $\mu \neq 0$	$(s)$ Charges at corners and mid-points of a rectangle. $M$ is the centre. $PQ$ is parallel to the longer sides.
	$(t)$ Charges on two coplanar,identical rings. $M$ is the mid-point between centres. $PQ$ is perpendicular to the line joining centres.

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As the temperature of the hot junction increases, the thermo $e.m.f.$

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