1. A point charge q with mass m is released from rest at the origin in...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
1. A point charge q with mass m is released from rest at the origin in an external magnetic field and an external electric field. Both of the external fields are uniform but the electric field points in the z-direction and the magnetic field points in the r-direction: = Eok B = Bo where Eo and Bo are constants. The electric field accelerates the point charge along the z-axis, but once the point charge has acquired a non-zero velocity along the 2-direction it will also experience a magnetic force. a) Using Newton's second law and the Lorentz force, show that the acceleration of the point charge in each direction must satisfy the following: dr dt where w is the cyclotron frequency d'y dt dz dt dz dt Eo dy dt qBo TIL and r(t), y(t), and z(t) give the position of the point charge along each axis as functions of time. b) The above equations are called coupled differential equations because they mix derivatives of y with derivatives of z (and vice versa). Show that these equations can be de-coupled by taking another time derivative. By de-coupled, I mean you should have an equation containing only derivatives of y with respect to t and an equation containing only derivatives of z with respect to t. c) The de-coupled equations are easier to solve, but let's not go through the actual steps. Instead I will just give you the general solution to the equations given in part a): x = Ct+C y=C3 cos wt + Ca sin wt + + C5 Ent Bo z = C4 cos wt - C3 sin wt + C6 where C, C2, C3, C4, C5, and Co are arbitrary constants. Show that these functions solve the equations from part a). You do NOT need to derive them, just plug them directly into the equations from part a). d) Recall that at t=0 the point charge is at rest and located at (x, y, z) = (0,0,0). Use these initial conditions to determine the values of the arbitrary constants. e) (Optional) Now try to sketch the motion of the charged particle. This can be tricky. It is helpful to take your expressions for y(t) and z(t) and notice that you can manipulate them into the equation for a circle, but one in which the center of the circle moves along the y-axis with constant speed. This is called cycloid motion and is identical to a wheel rolling at constant velocity. 1. A point charge q with mass m is released from rest at the origin in an external magnetic field and an external electric field. Both of the external fields are uniform but the electric field points in the z-direction and the magnetic field points in the r-direction: = Eok B = Bo where Eo and Bo are constants. The electric field accelerates the point charge along the z-axis, but once the point charge has acquired a non-zero velocity along the 2-direction it will also experience a magnetic force. a) Using Newton's second law and the Lorentz force, show that the acceleration of the point charge in each direction must satisfy the following: dr dt where w is the cyclotron frequency d'y dt dz dt dz dt Eo dy dt qBo TIL and r(t), y(t), and z(t) give the position of the point charge along each axis as functions of time. b) The above equations are called coupled differential equations because they mix derivatives of y with derivatives of z (and vice versa). Show that these equations can be de-coupled by taking another time derivative. By de-coupled, I mean you should have an equation containing only derivatives of y with respect to t and an equation containing only derivatives of z with respect to t. c) The de-coupled equations are easier to solve, but let's not go through the actual steps. Instead I will just give you the general solution to the equations given in part a): x = Ct+C y=C3 cos wt + Ca sin wt + + C5 Ent Bo z = C4 cos wt - C3 sin wt + C6 where C, C2, C3, C4, C5, and Co are arbitrary constants. Show that these functions solve the equations from part a). You do NOT need to derive them, just plug them directly into the equations from part a). d) Recall that at t=0 the point charge is at rest and located at (x, y, z) = (0,0,0). Use these initial conditions to determine the values of the arbitrary constants. e) (Optional) Now try to sketch the motion of the charged particle. This can be tricky. It is helpful to take your expressions for y(t) and z(t) and notice that you can manipulate them into the equation for a circle, but one in which the center of the circle moves along the y-axis with constant speed. This is called cycloid motion and is identical to a wheel rolling at constant velocity.
Expert Answer:
Answer rating: 100% (QA)
a Using Newtons second law and the Lorentz force we can write the equations of motion for the point charge in each direction In the xdirection m dxdt ... View the full answer
Related Book For
Fundamentals of Physics
ISBN: 978-0471758013
8th Extended edition
Authors: Jearl Walker, Halliday Resnick
Posted Date:
Students also viewed these physics questions
-
Hahndorf Ltd acquired 100% of the shares of Sarina Ltd on 1 July 2015 for S700,000, when the equity of Sarina Ltd consisted of: Share Capital General Reserve Retained Earnings $500,000 80,000 30,000...
-
Incorporated in 1990, Raju Diary Ltd is one of the leading manufacturers and marketers of diary-based branded foods in India. In the initial years, its operations were restricted only to the...
-
A 20-year, 4% quarterly coupon, $1,000 par value bond is selling for $1,075.31 with. Find its YTM in (a) APR and (b) EAR.
-
A company has net income of $ 265,000, a profit margin of 9.3 percent, and an accounts receivable balance of $ 145,300. Assuming 80 percent of sales are on credit, what is the companys days sales in...
-
A centrifugal radial water pump has the dimensions shown in Fig. P12.10. The volume rate of flow is \(0.25 \mathrm{ft}^{3} / \mathrm{s}\), and the absolute inlet velocity is directed radially...
-
Perform horizontal analysis of the data in the previous table to indicate any potential red flags regarding possible overstatement ofsales. FIVE-YEAR FINANCIAL DATA FOR COMPANY A YEAR I YEAR 2...
-
1.- Solve the following system of equations using Jacobi method, with epsilon=10-5 10x1 + 3x2+5x3=-3 X1-9x2 + 3x3 =5 -X1+22-15x3=8 2.- Solve the same system of equations using Gauss Seidel, with...
-
At the end of June, Rick Fontan decided to open his own computer service. Analyze the following transactions he completed by recording their effects in the expanded accounting equation. a. Invested...
-
A circuit has four inputs and one output. The inputs A3:0 represent a number from 0 to 15. The output P should be TRUE if the number is prime (0 and 1 are not prime, but 2, 3, 5, and so on, are...
-
A company reports net income of $ 8 , 90 0 for the year During the year, the company reports an $ 8 90 loss from retirement of notes and a gain of $ 1 , 42 0 from the sale of equipment Also,...
-
SCENARIO: Little Tummy Foods (LTF) is an Auckland-based company that manufactures "on-the-go" snacks, such as muffins, nut mixes, and muesli bars. Over the past years, the brand has gained...
-
A rope is spanned with no slack from the top of a 4 0 m high tower to the ground. The rope makes an angle of 2 0 degrees with the ground. What is the length of the rope? Please explain in detail how...
-
There are cultures that believe eye contact is disrespectful. Name at least one (1) of those cultures, its location in this world, and the reasons why eye contact is a sign of disrespect. Why do you...
-
What isNotice of appearance What is Time frame to serve the other party What is The party bringing the motion called What is The party responds to the motion What is Partial indemnity Request to...
-
Write a paper on the movie of your choice Start your paper by summarizing the plot of the movie and specifying the five concepts that the film demonstrates. This introduction should be brief - only...
-
Doorharmony Company makes doorbells. It has a weighted- average cost of capital of 5% and total assets of $ 5,900,000. Doorharmony has current liabilities of $ 750,000. Its operating income for the...
-
A four-person bobsled (total mass = 630 kg) comes down a straightaway at the start of a bobsled run. The straightaway is 80.0m long and is inclined at a constant angle of 10.2o with the horizontal...
-
A locomotive accelerates a 25-car train along a level track. Every car has a mass of 5.0 x 10 4 kg and is subject to a frictional force f = 250v, where the speed v is in meters per second and the...
-
A 1400 kg car moving at 5.3m/s is initially traveling north along the positive direction of a y axis. After completing a 90o right-hand turn rn 4.6 s, the inattentive operator drives into a tree,...
-
Two projects of equal value have been evaluated using a multifactor productivity measure that combined labor-hours at \($30\) per hour and overhead expense calculated as 20% of labor expense for the...
-
One road-paving crew of six people worked 8 hours to pave 25 miles of two-lane road, while a smaller crew of four people worked 12 hours to pave 15 miles of four- lane road. Which crew was more...
-
The Bankston fulfilment center of Universal Logistics is a large warehouse that packs and ships customer orders from client on-line retailers. Five employees have been nominated for a productivity...
Study smarter with the SolutionInn App