Question:
A proton of mass $ 1.67 \times 10^{-27}\, kg$ enters a uniform magnetic field $1\, T$ of at point A shown in figure with a speed of $10^7\, ms^{-1}$
The magnetic field is directed normal to the plane of paper downwards. The proton emerges out of the magnetic field at point $C$, then the distance
$AC$ and the value of angle $\theta $ will respectively be
A proton of mass $ 1.67 \times 10^{-27}\, kg$ enters a uniform magnetic field $1\, T$ of at point A shown in figure with a speed of $10^7\, ms^{-1}$
The magnetic field is directed normal to the plane of paper downwards. The proton emerges out of the magnetic field at point $C$, then the distance
$AC$ and the value of angle $\theta $ will respectively be
Updated On: Apr 15, 2024
- $0.7\, m ,\, 45^{\circ}$
- $0.7\, m ,\, 90^{\circ}$
- $0.14\, m ,\, 90^{\circ}$
- $0.14\, m ,\, 45^{\circ}$
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The Correct Option is D
Solution and Explanation
From the symmetry of figure, the angle $\theta=45^{\circ}$.
The path of moving proton in a normal magnetic field is circular. If $r$ is the radius of the circular path, then from the figure
$A C-2 r \cos 45^{\circ}-2 r \times \frac{1}{\sqrt{2}}-\sqrt{2} r$ ...(i)
As $B q v=\frac{m v^{2}}{r}$ or $ r=\frac{m v}{B q}$
$A C =\frac{\sqrt{2} m v}{B q}=\frac{\sqrt{2} \times 1.67 \times 10^{-27} \times 10^{7}}{1 \times 1.6 \times 10^{-19}}$
$=0.14\, m$
The path of moving proton in a normal magnetic field is circular. If $r$ is the radius of the circular path, then from the figure
$A C-2 r \cos 45^{\circ}-2 r \times \frac{1}{\sqrt{2}}-\sqrt{2} r$ ...(i)
As $B q v=\frac{m v^{2}}{r}$ or $ r=\frac{m v}{B q}$
$A C =\frac{\sqrt{2} m v}{B q}=\frac{\sqrt{2} \times 1.67 \times 10^{-27} \times 10^{7}}{1 \times 1.6 \times 10^{-19}}$
$=0.14\, m$
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Concepts Used:
Magnetic Field
The magnetic field is a field created by moving electric charges. It is a force field that exerts a force on materials such as iron when they are placed in its vicinity. Magnetic fields do not require a medium to propagate; they can even propagate in a vacuum. Magnetic field also referred to as a vector field, describes the magnetic influence on moving electric charges, magnetic materials, and electric currents.
A magnetic field can be presented in two ways.
- Magnetic Field Vector: The magnetic field is described mathematically as a vector field. This vector field can be plotted directly as a set of many vectors drawn on a grid. Each vector points in the direction that a compass would point and has length dependent on the strength of the magnetic force.
- Magnetic Field Lines: An alternative way to represent the information contained within a vector field is with the use of field lines. Here we dispense with the grid pattern and connect the vectors with smooth lines.
Properties of Magnetic Field Lines
- Magnetic field lines never cross each other
- The density of the field lines indicates the strength of the field
- Magnetic field lines always make closed-loops
- Magnetic field lines always emerge or start from the north pole and terminate at the south pole.