Summary
To determine the velocity of a charged particle moving through an electric field, you can use the equation v = E/B, where v is the velocity in meters per second, E is the magnitude of the electric field in Newtons per Coulomb, and B is the magnitude of the magnetic field in Tesla. This equation is derived from the requirement for undisturbed movement of the particle, where the forces acting on the particle must be balanced and at a right angle to each other. This guide will provide a detailed explanation of the underlying physics, formulas, examples, and numerical problems to help you master the concept of finding velocity from an electric field.
Understanding the Relationship between Electric Field, Magnetic Field, and Velocity
The motion of a charged particle in the presence of both an electric field and a magnetic field is governed by the Lorentz force equation, which states that the force acting on the particle is the vector sum of the electric force and the magnetic force. The Lorentz force equation is given by:
F = q(E + v × B)
where F is the Lorentz force, q is the charge of the particle, E is the electric field, v is the velocity of the particle, and B is the magnetic field.
For the particle to move undisturbed, the Lorentz force must be zero, which means that the electric force and the magnetic force must be balanced and at a right angle to each other. This condition can be expressed as:
qE = qvB
Rearranging this equation, we get the formula for the velocity of the charged particle:
v = E/B
This equation is the basis for finding the velocity of a charged particle from the given electric and magnetic fields.
Assumptions and Limitations
It is important to note that the equation v = E/B is valid only for particles moving at nonrelativistic speeds, where the effects of special relativity can be neglected. If the particle is moving at relativistic speeds, the calculations become more complex, and the formula needs to be modified to account for the relativistic effects.
Additionally, the equation assumes that the electric and magnetic fields are uniform and constant over the path of the particle. In real-world scenarios, the fields may vary, and the particle’s motion may be more complicated.
Examples and Numerical Problems
Example 1
Suppose an electric field has a magnitude of 10 N/C, and a magnetic field has a magnitude of 2 T. What is the velocity of a charged particle that can pass through the fields undisturbed?
Using the formula v = E/B, we can calculate the velocity:
v = E/B
v = 10 N/C / 2 T
v = 5 m/s
Therefore, the velocity of the charged particle that can pass through the fields undisturbed is 5 m/s.
Example 2
An electric field has a magnitude of 1 N/C, and a magnetic field has a magnitude of 4 T. What is the velocity of a charged particle that can pass through the fields undisturbed?
Using the formula v = E/B, we can calculate the velocity:
v = E/B
v = 1 N/C / 4 T
v = 0.25 m/s
Therefore, the velocity of the charged particle that can pass through the fields undisturbed is 0.25 m/s.
Numerical Problem 1
A charged particle with a mass of 2 × 10^-27 kg and a charge of 1.6 × 10^-19 C is moving through an electric field of 5 N/C and a magnetic field of 3 T. Calculate the velocity of the particle.
Given:
– Mass of the particle, m = 2 × 10^-27 kg
– Charge of the particle, q = 1.6 × 10^-19 C
– Electric field, E = 5 N/C
– Magnetic field, B = 3 T
Using the formula v = E/B, we can calculate the velocity:
v = E/B
v = 5 N/C / 3 T
v = 1.67 m/s
Therefore, the velocity of the charged particle is 1.67 m/s.
Numerical Problem 2
A proton (charge = 1.6 × 10^-19 C, mass = 1.67 × 10^-27 kg) is moving through an electric field of 2 N/C and a magnetic field of 1 T. Calculate the velocity of the proton.
Given:
– Charge of the proton, q = 1.6 × 10^-19 C
– Mass of the proton, m = 1.67 × 10^-27 kg
– Electric field, E = 2 N/C
– Magnetic field, B = 1 T
Using the formula v = E/B, we can calculate the velocity:
v = E/B
v = 2 N/C / 1 T
v = 2 m/s
Therefore, the velocity of the proton is 2 m/s.
Figures and Data Points
To further illustrate the relationship between electric field, magnetic field, and velocity, we can plot the velocity as a function of the electric and magnetic fields.
The plot shows that as the electric field increases, the velocity of the charged particle also increases, given a constant magnetic field. Conversely, as the magnetic field increases, the velocity of the charged particle decreases, given a constant electric field.
Additionally, we can provide a table of sample data points to demonstrate the relationship:
| Electric Field (N/C) | Magnetic Field (T) | Velocity (m/s) |
|---|---|---|
| 1 | 0.5 | 2 |
| 2 | 1 | 2 |
| 5 | 2 | 2.5 |
| 10 | 4 | 2.5 |
| 20 | 8 | 2.5 |
This table shows that as the electric field increases and the magnetic field increases, the velocity of the charged particle changes accordingly, following the formula v = E/B.
Conclusion
In this comprehensive guide, we have explored the concept of finding the velocity of a charged particle from an electric field. We have discussed the underlying physics, the formula v = E/B, and provided examples and numerical problems to help you understand and apply this concept. Remember that the equation is valid only for nonrelativistic speeds, and the fields must be uniform and constant over the particle’s path. By mastering this topic, you will be well-equipped to solve problems related to the motion of charged particles in electric and magnetic fields.
References
- Study.com: How to Determine the Velocity with which a Particle Can Move Undisturbed through a Pair of Uniform Magnetic and Electric Fields (https://study.com/skill/learn/how-to-determine-the-velocity-with-which-a-particle-can-move-undisturbed-through-a-given-pair-of-uniform-magnetic-and-electric-fields-explanation.html)
- Lumen Learning: Relativistic Energy (https://courses.lumenlearning.com/suny-physics/chapter/28-6-relativistic-energy/)
- Study.com: How to Determine Change in Velocity of a Charged Particle Moved through a Potential Difference (https://study.com/skill/learn/how-to-determine-change-in-velocity-of-a-charged-particle-moved-through-a-potential-difference-explanation.html)
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