A magnetic field of (10-4\(\hat{k}\))T exerts a force of (4\(\hat{i}\)-3\(\hat{j}\))x10-12N on a particle having a charge of 10-9 C. The speed of the particle is:
40 m/s
40\(\sqrt{2}\) m/s
50 m/s
50\(\sqrt{3}\) m/s
100\(\sqrt{2}\)m/s
The Correct Option is C
Approach Solution - 1
Given parameters:
- Magnetic field \( \vec{B} = 10^{-4} \hat{k} \, \text{T} \)
- Force \( \vec{F} = (4\hat{i} - 3\hat{j}) \times 10^{-12} \, \text{N} \)
- Charge \( q = 10^{-9} \, \text{C} \)
Lorentz force equation: \[ \vec{F} = q(\vec{v} \times \vec{B}) \]
Velocity components: \[ (4\hat{i} - 3\hat{j}) \times 10^{-12} = 10^{-9}(\vec{v} \times 10^{-4}\hat{k}) \] \[ \vec{v} \times \hat{k} = (4\hat{i} - 3\hat{j}) \times 10 \]
Solve for velocity: \[ \vec{v} = 40\hat{j} + 30\hat{i} \] \[ |\vec{v}| = \sqrt{30^2 + 40^2} = 50 \, \text{m/s} \]
Thus, the correct option is (C): 50 m/s.
Approach Solution -2
1. Recall the magnetic force equation:
The magnetic force (F) on a charged particle moving in a magnetic field (B) is given by:
\[\vec{F} = q(\vec{v} \times \vec{B})\]
where:
- q is the charge of the particle
- v is the velocity of the particle
- B is the magnetic field
2. Define the given information:
\[\vec{B} = 10^{-4}\hat{k} \, T\]
\[\vec{F} = (4\hat{i} - 3\hat{j}) \times 10^{-12} \, N\]
\[q = 10^{-9} \, C\]
3. Set up the equation and solve for the velocity:
Let \(\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}\). Substituting into the magnetic force equation:
\[(4\hat{i} - 3\hat{j}) \times 10^{-12} = 10^{-9}((v_x\hat{i} + v_y\hat{j} + v_z\hat{k}) \times 10^{-4}\hat{k})\]
Calculate the cross product:
\[(4\hat{i} - 3\hat{j}) \times 10^{-12} = 10^{-9} \times 10^{-4}(v_x(\hat{i} \times \hat{k}) + v_y(\hat{j} \times \hat{k}) + v_z(\hat{k} \times \hat{k}))\]
\[(4\hat{i} - 3\hat{j}) \times 10^{-12} = 10^{-13}(-v_x\hat{j} + v_y\hat{i})\]
Equate the components:
\[4 \times 10^{-12} = 10^{-13}v_y \implies v_y = 40 \, m/s\]
\[-3 \times 10^{-12} = -10^{-13}v_x \implies v_x = 30 \, m/s\]
The velocity vector is \(\vec{v} = 30\hat{i} + 40\hat{j}\) (the z-component is indeterminate as \(v_z\) disappears in the cross product).
4. Calculate the speed:
The speed is the magnitude of the velocity vector:
\[|\vec{v}| = \sqrt{v_x^2 + v_y^2} = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50 \, m/s\]
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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.