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AP C - UNIT 9 Magnetic Fields and Forces

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Magnetic Materials All magnetic materials have two poles –Labeled: North and South poles –Like poles repel each other and opposite poles attract.

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Magnetic Monopoles Unlike electrostatics, magnetic monopoles have never been detected.

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Modern View of Magnetism charges in motion (currents): Magnetism is associated with charges in motion (currents): –microscopic currents in the atoms of magnetic materials. –macroscopic currents in the windings of an electromagnet.

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Magnetic Field Similar to E-field, but where attraction and repulsion is occurring due to magnetism, either magnets or moving charges.

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Magnetic Field Lines

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The Earths Magnetic Field The spinning iron core of the earth produces a magnetic field.

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Magnetic Fields Magnetic field is a vector. It has direction and can be cancelled by another field.

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Magnetic Field Units The SI unit of magnetic field is the Tesla (T). Dimensional analysis: 1 T = 1 N·s/(C·m) = 1 V·s /m 2 Sometimes we use a unit called a Gauss (G): 1 T = 10 4 G The earths magnetic field is about 0.5 G.

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Magnetic Force on Moving Charges particle is moving A charged particle in a static magnetic field will experience a magnetic force only if the particle is moving. If a charge q with velocity v moves in a magnetic field B and v makes an angle w.r.t. B, then the magnitude of the force on the charge is:

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Finding the Direction of Magnetic Force The direction of the magnetic force is always perpendicular to both and F max occurs when is perpendicular to F = 0 when is parallel to

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Notation To represent the z-axis on an xy plane of paper. B out of the page B into the page

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Right Hand Rule Draw vectors v and B with their tails at the location of the charge q. Point fingers of right hand along velocity vector v. Curl fingers towards Magnetic field vector B. Thumb points in direction of magnetic force F on q, perpendicular to both v and B.

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Use RHR to find F on moving charged particle for each situation a through f

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Motion of Charges in B Fields If a charged particle is moving perpendicular to a uniform magnetic field, its trajectory will be a circle because the force F=qvB is always perpendicular to the motion, and therefore centripetal.

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If path of charge is not perpendicular to field where it enters field at some angle, charge will follow a spiral path (helix) which will spiral around the B-field. The component of velocity that is parallel to the magnetic field is unaffected. Its circlular motion will drift at a constant speed along the magnetic field The perp component of the velocity to B-field causes the particle to executes uniform circular motion perpendicular to the magnetic field.

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Velocity Selector A particle is accelerated through a potential difference where it then enters a magnetic and electric field. If the forces are balanced, the particle will move only horizontally between the plates (assuming negligible gravity)

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Mass Spectrometer The mass spectrometer utilizes the velocity selector and is an instrument which can measure the masses and relative concentrations of atoms and molecules. The radius of turn yields the mass of the particles.

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An ion is accelerated through a voltage of 1000V with a charge of 1.6x C after which it enters a chamber with a uniform B-field equal to 80mT. The ion follows a semi- circle path and strikes a photographic plate 1.62m from where it entered. Find its mass.

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Hall Effect If an electric current flows through a conductor in a magnetic field, the magnetic field exerts a force on the moving charge carriers which tends to push them to one side of the conductor. A buildup of charge at the sides of the conductor will balance this magnetic influence w/ E-field, producing a measurable voltage between the two sides of the conductor. The charge carries will then just pass through in a straight line.

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Force on a Wire Similar to the force on a moving charge in a B field, we have for a conductor of length L carrying a conventional current of I in a B field. The force experienced by the conductor is:

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A thin, massless, uniform rod with length 0.20m is attached to a frictionless hinge at point P. A horizontal spring (k = 4.8 N/m) connects the other end of the rod to a vertical wall. A uniform B-field equal to 0.34T is shown and a 6.5A current exists in the rod directed towards the hinge. How much energy is stored in the spring? B P I 53.1 o FSFS FMFM

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Torque on a current loop A wire loop is freely pivoting in a uniform B-field (+x) as shown. Each side is length a. a a B

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Electric Motor An electric motor converts electrical energy to mechanical energy – The mechanical energy is in the form of rotational kinetic energy When area vector of loop is parallel to field there is no torque After the loop moves a ½ turn, the current needs to switch direction to keep it rotating. Inertia will carry it past the edge (top of rotation) and at that moment if the current is switched, the loop keeps going. If the current isnt switched and it passes the edge, it will rotate the other way and get nowhere

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Biot Savart Law This law is seen as the magnetic equivalent of Coulomb's Law (brute force way of doing it vs Gauss). This finds B at a point P, a distance r from the differential element of current Id s. 0 = T m/A, magnetic permeability of free space Current makes B-fields

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Biot-Savart Law – Set-Up The magnetic field is at some point P The length element is The wire is carrying a steady current of I is the field created by the current in the length segment ds To find the total field, sum up the contributions from all the current elements I – The integral is over the entire current distribution

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B-Field due to a long straight wire along y-axis 0

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Example b a Consider wire bent into the shape shown above. Find B at P if current is flowing clockwise in wire. P I

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Magnetic Field from a Wire, RHR #3 The magnetic field lines from a current form circles around a straight wire with the direction given by another right hand rule. The magnitude of the magnetic field a distance r from a straight wire is given by (just proven with rigorous proof)

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Force between 2 current-carrying wires What happens when current as shown in 2 parallel wires? I1I1 I2I2 a

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Force between two parallel wires a

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Find B NET at point P P 45 o d R I1I1 I2I2 I 2 = I 1 45 o

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Loop of Current Consider a coil of radius R with current flowing CW. Find B at center of coil. X

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A loop of Current is RHR #4 Fingers are current direction and thumb is magnetic north pole

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I d I A current I flows in the positive y direction in an infinite wire; a current I also flows in the loop as shown in the diagram. What is the direction of the net force on the loop? (a) F = -x (b) F = 0 (c) F = +x (d) F = +y (e) F = -y

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B-field due to a current loop a distance x away frontal view of loop of current ds

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Amperes Law Amperes Law is to magnetic fields as Gauss Law was to electric fields. Both are used for high symmetry problems. Similar to drawing a Gaussian surface

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NOW apply Amperes Law to find B surrounding long straight current carrying wire ( already did this using Biot Savart ) Consider long wire with current I into page. r

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Magnetic field inside a wire Find B inside the wire with uniform current a distance r from the center where r < R

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B-field of a solenoid Solenoid is a series of loops of current. One end acts like a N & the other like a S-pole. Ideal solenoid is approached when turns are closely spaced and length is much greater than radius of turns. Current would flow from left to right across top of solenoid

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Derivation of B inside solenoid Cross sectional view of solenoid showing current into page on top, out of page on bottom

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Derivation of B inside solenoid

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Toroid - Solenoid shaped like a doughnut

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A current I flows in an infinite straight wire in the +z direction as shown. A concentric infinite cylinder of radius R carries current 2 I in the -z direction. x x x x x x x x 2I I a b x y a) What is the magnetic field B x at point a, just outside the cylinder as shown? Take CW as positive for B. c) B x (b) > 0(a) B x (b) < 0 c) B x (a) > 0a) B x (a) < 0b) B x (a) = 0 b) What is the magnetic field B x at point b, just inside the cylinder as shown? b) B x (b) = 0

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Two cylindrical conductors each carry current I into the page as shown. The conductor on the left is solid and has radius R=3a. The conductor on the right has a hole in the middle and carries current only between R=a and R=3a. 3a a II What is the relation between the magnetic field at R = 6a for the two cases (L=left, R=right)? a) B L (6a)< B R (6a) b) B L (6a)= B R (6a) c) B L (6a)> B R (6a)

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Two cylindrical conductors each carry current I into the page as shown. The conductor on the left is solid and has radius R=3a. The conductor on the right has a hole in the middle and carries current only between R=a and R=3a. 3a a II What is the relation between the magnetic field at R = 2a for the two cases (L=left, R=right)? a) B L (2a)< B R (2a) b) B L (2a)= B R (2a) c) B L (2a)> B R (2a)

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Consider a thin, infinitely large sheet of current that carries a linear current density, λ. The current is out of the page. Find B near the sheet. Recall a similar problem with infinitely charged sheet of charge density, σ.

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