Motion in a Uniform Electric Field Trajectory Analysis

Study the motion of a charge in a constant uniform electric field, deriving equations of motion, integrating for momentum and kinetic energy, using relativistic dynamics for velocity, and obtaining the trajectory in the XY plane. The trajectory is found to be a catenary shape when the velocity is much less than the speed of light.

• Motion
• Uniform Electric Field
• Trajectory Analysis
• Relativistic Dynamics
• Catenary

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1. Motion in a constant uniform electric field Section 20

2. Choose coordinates so that X is parallel to E. Choose inertial reference frame so that there is no motion of charge e along Z. Y Then, the motion of e is confined to the XY plane. E e X

3. To obtain the trajectory 1. Obtain equation of motion in terms of relativistic momentum.

4. 2. Integrate to obtain momentum vs. time.

5. 3. Obtain the kinetic energy kin as function of time

6. 4. Use equations of relativistic dynamics to obtain differential equation for velocity. Components of velocity

7. 5. Integrate to obtain coordinates as function of time Parametric equation x vs. t Parametric equation y vs t.

8. 6. Eliminate t to obtain the trajectory: x vs. y First express t in terms of y Then substitute into x(t) equation

9. v<<c

10. Trajectory is a Catenary