Magnetism Example Problems with Solutions PDF

Preview

Summary

Magnetism Example Problems with Solutions...

Description

Magnetism: Example Problems with Solutions Moving Charge in a Magnetic Field 1. An ionized helium atom has a mass of 6.6 x 10-27 kg is projected perpendicular into a magnetic field with a magnitude of 0.75 T with a speed of 4.4 x 105 m/s. The radius of its path is 0.012 m. a. Which circle in the diagram shows the path followed by the electron, the red or the blue? Solution The force that a magnetic field exerts on a moving charge:

F  q v B sinθ Particle entering a magnetic field with velocity perpendicular to the field: F  q v B The force:  Is directed in the radial direction toward the center of the circular path  Is perpendicular to the velocity of the particle (and the magnetic field)  Causes a centripetal acceleration  The path of the particle is a circle perpendicular to the direction of the magnetic field.  Since it is perpendicular to the velocity of the particle, it does not do work: it changes only the direction of the velocity of the particle not its speed.  Direction: apply RHR 1 (see figure) Ans. Red b. Determine whether the charge of this particle is +1e or +2 e.

m  6.6  10-27 kg B  0.75 T v  4.4  10 5 m/s R  0.012 m q ?

v2 mv  q  R RB  6.6 10-27 kg4.4 x 10 5 m/s  q 0.012 m0.75 T q 2 e q v B m

q  3.226  10-19 m

Ans: The charge of this particle is +2e c. How long does it take the particle to complete one revolution (hint: 𝑇 = 2πR v 2 π  0.012 m  T 4.4 x 10 5 m/s T





T  1.7  10 -7 s 1

𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑒𝑑 )? 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑠𝑝𝑒𝑒𝑑

Mass Spectrometry (see text book p. 881- 882) 2. A velocity selector in a mass spectrometer uses a 0.100-T magnetic field. a. What electric field strength is needed to select a speed of 4.00 106 m/s ? b. What is the voltage between the plates if they are separated by 1.00 cm? Solution a. The velocity selector has both an electric field and a magnetic field, perpendicular to one another, producing forces in opposite directions on the ions. Only those ions for which these forces cancel each other travel in a straight line into the next region. If the forces balance, then the electric force F = qE equals the magnetic force F = qvB , so that qE = qvB

E B    E vB E 4.00  10 6 m/s 0.100 T 

q E q v B  v 





E  4.00 10 V/m 5

b.





ΔV  E d  ΔV  4.00 105 V/m 0.01m ΔV  4.00  10 V

ΔV  4.00 kV

3

3. Triply charged uranium-235 and uranium-238 ions are being separated in a mass spectrometer. (The much rarer uranium-235 is used as reactor fuel.) The masses of the ions are 3.90 1025 kg and 3.95 1025 kg , respectively, and they travel at 3.00 105 m/s in a 0.250T field. What is the separation between their paths when they hit a target after traversing a semicircle? (b) Discuss whether this distance between their paths seems to be big enough to be practical in the separation of uranium-235 from uranium-238. d  2 r - 2 r 1

Solution q  3 1.6 10 19 C





m 1  3.90 10-25 kg m 2  3.95  10 -25 kg v  3.00 10 5 m/s B  0.250 T d  ?

2

F q v B m v2 r m v2 mv r  qvB r qB

F

d  2 r2 - 2 r1  2

2 r1

2 r2

m v m2 v -2 1 qB qB

d 

2 v  m2 - m1 qB

Δd 

2 3.00  10 5 3.95  10 -25 kg - 3.90  10 -25 kg 3 1.6  10 -19 C 0.250 T 





 



2



d  2.5  10-2 m

Magnetic Force on a Current Carrying Conductor 4. A long, straight wire carries a 6.0-A current that is directed in the positive x direction. When a uniform magnetic field is applied perpendicular to a 3.0-m segment of the wire, the magnetic force on the segment is 0.36 N, directed in the negative y direction, as shown. What are the magnitude and direction of the magnetic field? I6A F Fm  B I L B  m L 3m IL Fm  0.36 N  0.36 N  B 6 A 3 m  B ? B  0.02 T out of the paper Magnetic Fields Produced by Currents 5. Two long, straight wires are perpendicular to the plane of the paper as shown in the drawing. The wires carry currents of I1 = 3 A and I2 = 5 A in the direction indicated (out of the page). a. Draw the direction of the magnetic field due to current I1 at a point A midway between the two wires. y

b. Draw the direction of the magnetic field due to current I2 at point A. Find its magnitude.

B1

c. Find the magnitude and direction of the magnetic field at a point A midway between the wires.

A

r1  r1  0.10 m I1  3 A I 1  5 A B2

BA  ? B A  B1  B 2  BA 

μ 0 I 1 μ0 I 2 2 π r1 2 π r2

μ 0  I1 I2   -  2 π  r1 r2 

 4 π 10



T m/A  3.00 A 5.00 A    2 0.10   0.10 -7

BA 

BA   4 10-6 T The direction: the net magnetic field is in the negative y direction (downward).

3

x

Torque on a Current Loop 6. A circular coil consists of 5 loops each of diameter 1.0 m. The coil is placed in an external magnetic field of 0.5 T. When the coil carries a current of 4.0 A, a torque of magnitude 3.93 N-m acts on it. Determine the magnetic moment of the coil (µ). What is the angle between the normal to the plane of the coil and the direction of the magnetic field? r  0.5 m μ  NIA θ N 5 2 2 μ  54 A π0.5 m   15.7 A m B  0.500 T τ I 4 A τ  μ B sin θ  θ  sin -1 μB τ  3.93 N m 3.93 N m θ? θ  sin -1 15.7 A m 2 0.500 T  μ?



θ  30





Magnetic Force between Two Parallel Conductors 7. Two parallel wires are separated by 0.06 m, each carrying 3 A of current in the same direction. What is the magnitude of the force per unit length between the wires? Is the force attractive or repulsive?

F μ0 I1 I2  L 2πd 2 F 4 π 10-7 T  m/A 3 A   L 2 π 0.06 m F  3  10 -5 N/m L The force is attractive.



Fm



4...