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Calculating magnetic force: earth’s magnetic field on a charged glass rod

With the exception of compasses, you seldom see or personally experience forces due to the Earth’s small magnetic field. To illustrate this, suppose that in a physics lab you rub a glass rod with silk, placing a 20-nC positive charge on it. Calculate the force on the rod due to the Earth’s magnetic field, if you throw it with a horizontal velocity of 10 m/s due west in a place where the Earth’s field is due north parallel to the ground. (The direction of the force is determined with right hand rule 1 as shown in [link] .)

The effects of the Earth’s magnetic field on moving charges. Figure a shows a positive charge with a velocity vector due west, a magnetic field line B oriented due north, and a magnetic force vector F straight down. Figure b shows the right hand facing down, with the fingers pointing north with B, the thumb pointing west with v, and force down away from the hand.
A positively charged object moving due west in a region where the Earth’s magnetic field is due north experiences a force that is straight down as shown. A negative charge moving in the same direction would feel a force straight up.

Strategy

We are given the charge, its velocity, and the magnetic field strength and direction. We can thus use the equation F = qvB sin θ size 12{F= ital "qvB""sin"θ} {} to find the force.

Solution

The magnetic force is

F = qvb sin θ . size 12{F= ital "qvb""sin"θ} {}

We see that sin θ = 1 size 12{"sin"θ=1} {} , since the angle between the velocity and the direction of the field is 90º size 12{"90" rSup { size 8{ circ } } } {} . Entering the other given quantities yields

F = 20 × 10 –9 C 10 m/s 5 × 10 –5 T = 1 × 10 –11 C m/s N C m/s = 1 × 10 –11 N. alignl { stack { size 12{F= left ("20" times "10" rSup { size 8{ - 9 } } `C right ) left ("10"`"m/s" right ) left (5 times "10" rSup { size 8{ - 5} } `T right )} {} #" "=1 times "10" rSup { size 8{ - "11"} } ` left (C cdot "m/s" right ) left ( { {N} over {C cdot "m/s"} } right )=1 times "10" rSup { size 8{ - "11"} } `N "." {} } } {}

Discussion

This force is completely negligible on any macroscopic object, consistent with experience. (It is calculated to only one digit, since the Earth’s field varies with location and is given to only one digit.) The Earth’s magnetic field, however, does produce very important effects, particularly on submicroscopic particles. Some of these are explored in Force on a Moving Charge in a Magnetic Field: Examples and Applications .

Section summary

  • Magnetic fields exert a force on a moving charge q , the magnitude of which is
    F = qvB sin θ , size 12{F= ital "qvB""sin"θ} {}
    where θ size 12{θ} {} is the angle between the directions of v size 12{v} {} and B size 12{B} {} .
  • The SI unit for magnetic field strength B size 12{B} {} is the tesla (T), which is related to other units by
    1 T = 1 N C m/s = 1 N A m .
  • The direction of the force on a moving charge is given by right hand rule 1 (RHR-1): Point the thumb of the right hand in the direction of v size 12{v} {} , the fingers in the direction of B size 12{B} {} , and a perpendicular to the palm points in the direction of F size 12{F} {} .
  • The force is perpendicular to the plane formed by v and B size 12{B} {} . Since the force is zero if v size 12{v} {} is parallel to B size 12{B} {} , charged particles often follow magnetic field lines rather than cross them.

Conceptual questions

If a charged particle moves in a straight line through some region of space, can you say that the magnetic field in that region is necessarily zero?

Problems&Exercises

What is the direction of the magnetic force on a positive charge that moves as shown in each of the six cases shown in [link] ?

figure a shows magnetic field line direction symbols with solid circles labeled B out; a velocity vector points down; figure b shows B vectors pointing right and v vector pointing up; figure c shows B in and v to the right; figure d shows B vector pointing right and v vector pointing left; figure e shows B vectors up and v vector into the page; figure f shows B vectors pointing left and v vectors out of the page

(a) Left (West)

(b) Into the page

(c) Up (North)

(d) No force

(e) Right (East)

(f) Down (South)

Repeat [link] for a negative charge.

What is the direction of the velocity of a negative charge that experiences the magnetic force shown in each of the three cases in [link] , assuming it moves perpendicular to B ? size 12{B?} {}

Figure a shows the force vector pointing up and B out of the page. Figure b shows the F vector pointing up and the B vector pointing to the right. Figure c shows the F vector pointing to the left and the B vector pointing into the page.

(a) East (right)

(b) Into page

(c) South (down)

Repeat [link] for a positive charge.

What is the direction of the magnetic field that produces the magnetic force on a positive charge as shown in each of the three cases in the figure below, assuming B size 12{B} {} is perpendicular to v size 12{v} {} ?

Figure a shows a force vector pointing toward the left and a velocity vector pointing up. Figure b shows the force vector pointing into the page and the velocity vector pointing down. Figure c shows the force vector pointing up and the velocity vector pointing to the left.

(a) Into page

(b) West (left)

(c) Out of page

Repeat [link] for a negative charge.

What is the maximum force on an aluminum rod with a 0 . 100 -μC size 12{0 "." "100""-μC"} {} charge that you pass between the poles of a 1.50-T permanent magnet at a speed of 5.00 m/s? In what direction is the force?

7 . 50 × 10 7 N size 12{7 "." "50" times "10" rSup { size 8{ - 7} } " N"} {} perpendicular to both the magnetic field lines and the velocity

(a) Aircraft sometimes acquire small static charges. Suppose a supersonic jet has a 0 . 500 -μC size 12{0 "." "500""-μC"} {} charge and flies due west at a speed of 660 m/s over the Earth’s south magnetic pole, where the 8 . 00 × 10 5 -T size 12{8 "." "00" times "10" rSup { size 8{ - 5} } "-T"} {} magnetic field points straight up. What are the direction and the magnitude of the magnetic force on the plane? (b) Discuss whether the value obtained in part (a) implies this is a significant or negligible effect.

(a) A cosmic ray proton moving toward the Earth at 5.00 × 10 7 m/s size 12{5 "." "00" times "10" rSup { size 8{7} } `"m/s"} {} experiences a magnetic force of 1 . 70 × 10 16 N size 12{1 "." "70" times "10" rSup { size 8{ - "16"} } `N} {} . What is the strength of the magnetic field if there is a 45º size 12{"45" rSup { size 8{ circ } } } {} angle between it and the proton’s velocity? (b) Is the value obtained in part (a) consistent with the known strength of the Earth’s magnetic field on its surface? Discuss.

(a) 3 . 01 × 10 5 T size 12{3 "." "01" times "10" rSup { size 8{ - 5} } " T"} {}

(b) This is slightly less then the magnetic field strength of 5 × 10 5 T size 12{5 times "10" rSup { size 8{ - 5} } `T} {} at the surface of the Earth, so it is consistent.

An electron moving at 4 . 00 × 10 3 m/s size 12{4 "." "00" times "10" rSup { size 8{3} } `"m/s"} {} in a 1.25-T magnetic field experiences a magnetic force of 1 . 40 × 10 16 N size 12{1 "." "40" times "10" rSup { size 8{ - "16"} } `N} {} . What angle does the velocity of the electron make with the magnetic field? There are two answers.

(a) A physicist performing a sensitive measurement wants to limit the magnetic force on a moving charge in her equipment to less than 1 . 00 × 10 12 N size 12{1 "." "00" times "10" rSup { size 8{ - "12"} } `N} {} . What is the greatest the charge can be if it moves at a maximum speed of 30.0 m/s in the Earth’s field? (b) Discuss whether it would be difficult to limit the charge to less than the value found in (a) by comparing it with typical static electricity and noting that static is often absent.

(a) 6 . 67 × 10 10 C (taking the Earth’s field to be 5 . 00 × 10 5 T size 12{5 "." "00" times "10" rSup { size 8{ - 5} } " T"} {} )

(b) Less than typical static, therefore difficult

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Source:  OpenStax, General physics ii phy2202ca. OpenStax CNX. Jul 05, 2013 Download for free at http://legacy.cnx.org/content/col11538/1.2
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