Problem 2: A charged particle of mass m and positive charge q moves in uniform electric and magnetic fields E = Eŷ and B = B2. Suppose the particle starts at the origin with initial velocity vox. (a) Write down the (vector) equation of motion for the particle and resolve it into its three components. Show that the motion remains in the plane z = 0. (b) Prove that there is a unique value of vo, known as the drift speed Vdr, for which the charge moves undeflected through the fields, and write an expression for vår in terms of the given quantities. (c) Solve the equations of motion to find v(t), for arbitrary initial velocity vo in the x direction. It may be helpful to write your answer using var. [Hint: try variable sub- stitution to simplify your differential equations. The drift speed seems important, so shift to a reference frame that moves at that speed.]
Problem 2: A charged particle of mass m and positive charge q moves in uniform electric and magnetic fields E = Eŷ and B = B2. Suppose the particle starts at the origin with initial velocity vox. (a) Write down the (vector) equation of motion for the particle and resolve it into its three components. Show that the motion remains in the plane z = 0. (b) Prove that there is a unique value of vo, known as the drift speed Vdr, for which the charge moves undeflected through the fields, and write an expression for vår in terms of the given quantities. (c) Solve the equations of motion to find v(t), for arbitrary initial velocity vo in the x direction. It may be helpful to write your answer using var. [Hint: try variable sub- stitution to simplify your differential equations. The drift speed seems important, so shift to a reference frame that moves at that speed.]
Related questions
Question
![**Problem 2**: A charged particle of mass \( m \) and positive charge \( q \) moves in uniform electric and magnetic fields \( \mathbf{E} = E\hat{\mathbf{y}} \) and \( \mathbf{B} = B\hat{\mathbf{z}} \). Suppose the particle starts at the origin with initial velocity \( v_0\hat{\mathbf{x}} \).
(a) Write down the (vector) equation of motion for the particle and resolve it into its three components. Show that the motion remains in the plane \( z = 0 \).
(b) Prove that there is a unique value of \( v_0 \), known as the drift speed \( v_{dr} \), for which the charge moves undeflected through the fields, and write an expression for \( v_{dr} \) in terms of the given quantities.
(c) Solve the equations of motion to find \( \mathbf{v}(t) \), for arbitrary initial velocity \( v_0 \) in the \( x \) direction. It may be helpful to write your answer using \( v_{dr} \). [Hint: Try variable substitution to simplify your differential equations. The drift speed seems important, so shift to a reference frame that moves at that speed.]
(d) Integrate to find \( \mathbf{r}(t) \), and plot the trajectory for several different and representative values of the ratio \( v_0/v_{dr} \).
For each part, check your result for dimensional consistency and limiting-case behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29c74d06-0f3b-4eb2-9c9d-dbbc1918002c%2F13ed64ce-0122-462c-92c0-d92d15201a30%2Fpft689i_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 2**: A charged particle of mass \( m \) and positive charge \( q \) moves in uniform electric and magnetic fields \( \mathbf{E} = E\hat{\mathbf{y}} \) and \( \mathbf{B} = B\hat{\mathbf{z}} \). Suppose the particle starts at the origin with initial velocity \( v_0\hat{\mathbf{x}} \).
(a) Write down the (vector) equation of motion for the particle and resolve it into its three components. Show that the motion remains in the plane \( z = 0 \).
(b) Prove that there is a unique value of \( v_0 \), known as the drift speed \( v_{dr} \), for which the charge moves undeflected through the fields, and write an expression for \( v_{dr} \) in terms of the given quantities.
(c) Solve the equations of motion to find \( \mathbf{v}(t) \), for arbitrary initial velocity \( v_0 \) in the \( x \) direction. It may be helpful to write your answer using \( v_{dr} \). [Hint: Try variable substitution to simplify your differential equations. The drift speed seems important, so shift to a reference frame that moves at that speed.]
(d) Integrate to find \( \mathbf{r}(t) \), and plot the trajectory for several different and representative values of the ratio \( v_0/v_{dr} \).
For each part, check your result for dimensional consistency and limiting-case behavior.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 7 steps with 5 images
