A particle of mass $M$ and charge $Q$ moving with velocity $\mathop v\limits^ \to $ describes a circular path of radius $R$ when subjected to a uniform transverse magnetic field of induction $B$. The work done by the field when the particle completes one full circle is
A particle of mass $M$ and charge $Q$ moving with velocity $\mathop v\limits^ \to $ describes a circular path of radius $R$ when subjected to a uniform transverse magnetic field of induction $B$. The work done by the field when the particle completes one full circle is
- [AIEEE 2003]
- A
$B\,Qv\,2\pi \,R$
- B
$\left( {\frac{{M{v^2}}}{R}} \right)\,2\pi R$
- C
Zero
- D
$BQ2\pi R$
Similar Questions
A charged particle (electron or proton) is introduced at the origin $(x=0, y=0, z=0)$ with a given initial velocity $\overrightarrow{\mathrm{v}}$. A uniform electric field $\overrightarrow{\mathrm{E}}$ and magnetic field $\vec{B}$ are given in columns $1,2$ and $3$ , respectively. The quantities $E_0, B_0$ are positive in magnitude.
column $I$
column $II$
column $III$
$(I)$ Electron with $\overrightarrow{\mathrm{v}}=2 \frac{\mathrm{E}_0}{\mathrm{~B}_0} \hat{\mathrm{x}}$
$(i)$ $\overrightarrow{\mathrm{E}}=\mathrm{E}_0^2 \hat{\mathrm{Z}}$
$(P)$ $\overrightarrow{\mathrm{B}}=-\mathrm{B}_0 \hat{\mathrm{x}}$
$(II)$ Electron with $\overrightarrow{\mathrm{v}}=\frac{\mathrm{E}_0}{\mathrm{~B}_0} \hat{\mathrm{y}}$
$(ii)$ $\overrightarrow{\mathrm{E}}=-\mathrm{E}_0 \hat{\mathrm{y}}$
$(Q)$ $\overrightarrow{\mathrm{B}}=\mathrm{B}_0 \hat{\mathrm{x}}$
$(III)$ Proton with $\overrightarrow{\mathrm{v}}=0$
$(iii)$ $\overrightarrow{\mathrm{E}}=-\mathrm{E}_0 \hat{\mathrm{x}}$
$(R)$ $\overrightarrow{\mathrm{B}}=\mathrm{B}_0 \hat{\mathrm{y}}$
$(IV)$ Proton with $\overrightarrow{\mathrm{v}}=2 \frac{\mathrm{E}_0}{\mathrm{~B}_0} \hat{\mathrm{x}}$
$(iv)$ $\overrightarrow{\mathrm{E}}=\mathrm{E}_0 \hat{\mathrm{x}}$
$(S)$ $\overrightarrow{\mathrm{B}}=\mathrm{B}_0 \hat{\mathrm{z}}$
($1$) In which case will the particle move in a straight line with constant velocity?
$[A] (II) (iii) (S)$ $[B] (IV) (i) (S)$ $[C] (III) (ii) (R)$ $[D] (III) (iii) (P)$
($2$) In which case will the particle describe a helical path with axis along the positive $z$ direction?
$[A] (II) (ii) (R)$ $[B] (IV) (ii) (R)$ $[C] (IV) (i) (S)$ $[D] (III) (iii)(P)$
($3$) In which case would be particle move in a straight line along the negative direction of y-axis (i.e., more along $-\hat{y}$ )?
$[A] (IV) (ii) (S)$ $[B] (III) (ii) (P)$ $[C]$ (II) (iii) $(Q)$ $[D] (III) (ii) (R)$
A charged particle (electron or proton) is introduced at the origin $(x=0, y=0, z=0)$ with a given initial velocity $\overrightarrow{\mathrm{v}}$. A uniform electric field $\overrightarrow{\mathrm{E}}$ and magnetic field $\vec{B}$ are given in columns $1,2$ and $3$ , respectively. The quantities $E_0, B_0$ are positive in magnitude.
|
column $I$ |
column $II$ | column $III$ |
| $(I)$ Electron with $\overrightarrow{\mathrm{v}}=2 \frac{\mathrm{E}_0}{\mathrm{~B}_0} \hat{\mathrm{x}}$ | $(i)$ $\overrightarrow{\mathrm{E}}=\mathrm{E}_0^2 \hat{\mathrm{Z}}$ | $(P)$ $\overrightarrow{\mathrm{B}}=-\mathrm{B}_0 \hat{\mathrm{x}}$ |
| $(II)$ Electron with $\overrightarrow{\mathrm{v}}=\frac{\mathrm{E}_0}{\mathrm{~B}_0} \hat{\mathrm{y}}$ | $(ii)$ $\overrightarrow{\mathrm{E}}=-\mathrm{E}_0 \hat{\mathrm{y}}$ | $(Q)$ $\overrightarrow{\mathrm{B}}=\mathrm{B}_0 \hat{\mathrm{x}}$ |
| $(III)$ Proton with $\overrightarrow{\mathrm{v}}=0$ | $(iii)$ $\overrightarrow{\mathrm{E}}=-\mathrm{E}_0 \hat{\mathrm{x}}$ | $(R)$ $\overrightarrow{\mathrm{B}}=\mathrm{B}_0 \hat{\mathrm{y}}$ |
| $(IV)$ Proton with $\overrightarrow{\mathrm{v}}=2 \frac{\mathrm{E}_0}{\mathrm{~B}_0} \hat{\mathrm{x}}$ | $(iv)$ $\overrightarrow{\mathrm{E}}=\mathrm{E}_0 \hat{\mathrm{x}}$ | $(S)$ $\overrightarrow{\mathrm{B}}=\mathrm{B}_0 \hat{\mathrm{z}}$ |
($1$) In which case will the particle move in a straight line with constant velocity?
$[A] (II) (iii) (S)$ $[B] (IV) (i) (S)$ $[C] (III) (ii) (R)$ $[D] (III) (iii) (P)$
($2$) In which case will the particle describe a helical path with axis along the positive $z$ direction?
$[A] (II) (ii) (R)$ $[B] (IV) (ii) (R)$ $[C] (IV) (i) (S)$ $[D] (III) (iii)(P)$
($3$) In which case would be particle move in a straight line along the negative direction of y-axis (i.e., more along $-\hat{y}$ )?
$[A] (IV) (ii) (S)$ $[B] (III) (ii) (P)$ $[C]$ (II) (iii) $(Q)$ $[D] (III) (ii) (R)$
- [IIT 2017]
A small block of mass $m$, having charge $q$ is placed on frictionless inclined plane making an angle $\theta$ with the horizontal. There exists a uniform magnetic field $B$ parallel to the inclined plane but perpendicular to the length of spring. If $m$ is slightly pulled on the inclined in downward direction and released, the time period of oscillation will be (assume that the block does not leave contact with the plane)
A small block of mass $m$, having charge $q$ is placed on frictionless inclined plane making an angle $\theta$ with the horizontal. There exists a uniform magnetic field $B$ parallel to the inclined plane but perpendicular to the length of spring. If $m$ is slightly pulled on the inclined in downward direction and released, the time period of oscillation will be (assume that the block does not leave contact with the plane)
The time period of a charged particle undergoing a circular motion in a uniform magnetic field is independent of its
The time period of a charged particle undergoing a circular motion in a uniform magnetic field is independent of its
- [AIEEE 2002]
A particle with ${10^{ - 11}}\,coulomb$ of charge and ${10^{ - 7}}\,kg$ mass is moving with a velocity of ${10^8}\,m/s$ along the $y$-axis. A uniform static magnetic field $B = 0.5\,Tesla$ is acting along the $x$-direction. The force on the particle is
A particle with ${10^{ - 11}}\,coulomb$ of charge and ${10^{ - 7}}\,kg$ mass is moving with a velocity of ${10^8}\,m/s$ along the $y$-axis. A uniform static magnetic field $B = 0.5\,Tesla$ is acting along the $x$-direction. The force on the particle is
A charged particle is projected in a plane perpendicular to a uniform magnetic field. The area bounded by the path described by the particle is proportional to
A charged particle is projected in a plane perpendicular to a uniform magnetic field. The area bounded by the path described by the particle is proportional to