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Magnetic Flux Density Magnetic Flux density: A Magnetic flux lines are continuous and closed.Magnetic flux lines are continuous and closed. Direction is that of the B vector at any point.Direction is that of the B vector at any point. Flux lines are NOT in direction of force but.Flux lines are NOT in direction of force but. When area A is perpendicular to flux: The unit of flux density is the Weber per square meter.

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Calculating Flux Density When Area is Not Perpendicular The flux penetrating the area A when the normal vector n makes an angle of with the B-field is: The angle is the complement of the angle a that the plane of the area makes with the B field. (Cos = Sin The angle is the complement of the angle a that the plane of the area makes with the B field. (Cos = Sin n A B

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Origin of Magnetic Fields Recall that the strength of an electric field E was defined as the electric force per unit charge. Since no isolated magnetic pole has ever been found, we cant define the magnetic field B in terms of the magnetic force per unit north pole. We will see instead that magnetic fields result from charges in motionnot from stationary charge or poles. This fact will be covered later. + E + B v v

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Magnetic Force on Moving Charge NS B N Imagine a tube that projects charge +q with velocity v into perpendicular B field. Upward magnetic force F on charge moving in B field. vF Experiment shows: Each of the following results in a greater magnetic force F: an increase in velocity v, an increase in charge q, and a larger magnetic field B.

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Direction of Magnetic Force B vF NSN The right hand rule: With a flat right hand, point thumb in direction of velocity v, fingers in direction of B field. The flat hand pushes in the direction of force F. The force is greatest when the velocity v is perpendicular to the B field. The deflection decreases to zero for parallel motion. B v F

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Force and Angle of Path SNNS NN SNN Deflection force greatest when path perpendicular to field. Least at parallel. B v F v sin v sin v

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Definition of B-field Experimental observations show the following: By choosing appropriate units for the constant of proportionality, we can now define the B-field as: Magnetic Field Intensity B: A magnetic field intensity of one tesla (T) exists in a region of space where a charge of one coulomb (C) moving at 1 m/s perpendicular to the B-field will experience a force of one newton (N).

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Example 1. A 2-nC charge is projected with velocity 5 x 10 4 m/s at an angle of 30 0 with a 3 mT magnetic field as shown. What are the magnitude and direction of the resulting force? v sin v sin v B v F Draw a rough sketch. q = 2 x C v = 5 x 10 4 m/s B = 3 x T = 30 0 Using right-hand rule, the force is seen to be upward. Resultant Magnetic Force: F = 1.50 x N, upward B

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Forces on Negative Charges Forces on negative charges are opposite to those on positive charges. The force on the negative charge requires a left-hand rule to show downward force F. NSNNSN B v F Right-hand rule for positive q F Bv Left-hand rule for negative q

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Indicating Direction of B-fields One way of indicating the directions of fields perpen- dicular to a plane is to use crosses X and dots One way of indicating the directions of fields perpen- dicular to a plane is to use crosses X and dots : X X X X X X X X X X X X X X X X A field directed into the paper is denoted by a cross X like the tail feathers of an arrow. A field directed out of the paper is denoted by a dot like the front tip end of an arrow.

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Practice With Directions: X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X What is the direction of the force F on the charge in each of the examples described below? -v - v +v v + UpF LeftF F Right UpF negative q

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Crossed E and B Fields The motion of charged particles, such as electrons, can be controlled by combined electric and magnetic fields. x x x x + - e-e- v Note: F E on electron is upward and opposite E-field. But, F B on electron is down (left-hand rule). Zero deflection when F B = F E B v FEFEFEFE Ee-e- - Bv FBFBFBFB -

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The Velocity Selector This device uses crossed fields to select only those velocities for which F B = F E. (Verify directions for +q) x x x x + - +q v Source of +q Velocity selector When F B = F E : By adjusting the E and/or B-fields, a person can select only those ions with the desired velocity.

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Example 2. A lithium ion, q = +1.6 x C, is projected through a velocity selector where B = 20 mT. The E-field is adjusted to select a velocity of 1.5 x 10 6 m/s. What is the electric field E? x x x x + - +q v Source of +qV E = vB E = (1.5 x 10 6 m/s)(20 x T); E = 3.00 x 10 4 V/m

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Circular Motion in B-field The magnetic force F on a moving charge is always perpendicular to its velocity v. Thus, a charge moving in a B-field will experience a centripetal force. X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Centripetal F c = F B R FcFcFcFc The radius of path is:

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Mass Spectrometer +q R + - x x x x x x x x x x x x x x x x x x x x x Photographic plate m1m1 m2m2 slit Ions passed through a velocity selector at known velocity emerge into a magnetic field as shown. The radius is: The mass is found by measuring the radius R:

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Example 3. A Neon ion, q = 1.6 x C, follows a path of radius 7.28 cm. Upper and lower B = 0.5 T and E = 1000 V/m. What is its mass? v = 2000 m/s m = 2.91 x kg +q R + - x x x x Photographic plate m slit x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

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Summary NSN B v F Right-hand rule for positive q NSN F Bv Left-hand rule for negative q The direction of forces on a charge moving in an electric field can be determined by the right-hand rule for positive charges and by the left-hand rule for negative charges.

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Summary (Continued) B v F v sin v sin v For a charge moving in a B-field, the magnitude of the force is given by: F = qvB sin

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Summary (Continued) x x x x + - +q+q v V The velocity selector: +q R + - x x x x m slit x x x x x x x x x x x x x The mass spectrometer:

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