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Magnetic Fields AP Physics C Montwood High School R. Casao

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Definition & Properties of the Magnetic Field The electric field E at a point in space is defined at the electric force per unit charge acting on a test charge placed at that point. The gravitational field g at a point in space is the gravitational force per unit mass acting on a test mass. The magnetic field B at some point in space is the magnetic force that would be exerted on a charged particle moving with a velocity v.

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Properties of the Magnetic Force on a Charge Moving in a Magnetic Field B The magnetic force F mag is proportional to the charge q and the speed v of the particle. The magnitude and direction of the magnetic force depend on the velocity of the particle and on the magnitude and direction of the magnetic field. When a charged particle moves in a direction parallel to the magnetic field vector, the magnetic force F mag on the charge is zero.

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Properties of the Magnetic Force on a Charge Moving in a Magnetic Field B When the velocity vector makes an angle with the magnetic field, the magnetic force acts in a direction perpendicular to both v and B; that is, F mag is perpendicular to the plane formed by v and B. The magnetic force on a positive charge is in the direction opposite the direction of the force on a negative charge moving in the same direction. If the velocity vector makes an angle with the magnetic field, the magnitude of the magnetic force is proportional to sin .

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Properties of the Magnetic Field Equation: –where the direction of the magnetic force is in the direction of v x B, which by definition of the cross product, is perpendicular to both v and B.

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Right-Hand Rule The right-hand rule is used to determine the direction of the cross product v X B. Figure a: the fingers of the right hand are placed in the direction of the velocity v. The palm of the right hand is placed in the direction of the magnetic field B. The fingers of the right hand are curled through angle from v towards B. The thumb points in the direction of the resultant vector v x B, or the magnetic force F mag.

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Right-Hand Rule Figure b: If charge q is positive, then the direction of the resultant vector v x B, or the magnetic force F mag, is in the direction of the thumb of the right hand (upwards as in the figure). Figure c: If charge q is negative, then the direction of the resultant vector v x B, or the magnetic force F mag, is in the oppposite direction of the thumb of the right hand (downwards).

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Cross-Products Whenever you see a cross-product, such as v X B: –the fingers of the hand will always point in the direction of the first variable. –the palm of the hand will always point in the direction of the second variable. –the thumb will always point in the direction of the resultant vector. All three vectors are at right angles to each other.

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Casao’s Version of the Right-Hand Rule I use the right hand as described for positive charges. No change here. I use the left hand for negative charges. –For the v X B cross product using a negative charge: –Point the fingers of the left hand in the direction of v. –Point the palm of the left hand in the direction of B. –The thumb of the left hand points in the direction of the resultant vector, v X B, or the magnetic force F mag.

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Casao’s Version of the Right-Hand Rule

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Equation The magnitude of the magnetic force: F mag = q·v·B·sin where is the angle between v and B. F mag = 0 N when v is parallel to B (sin 0 = 0). F mag is maximum when v is perpendicular to B (sin 90 = 1).

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Differences Between E and B The magnetic field is defined in terms of a sideways force acting on a moving charged particle. Differences between electric field E and the magnetic field B: –E is always in the direction of the electric field; B is perpendicular to the magnetic field. –E acts on a charged particle independent of the particle’s velocity (the charge can be stationary); B acts on a charged particle only when the particle is in motion.

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Differences Between E and B –E does work in displacing a charged particle because the direction of motion is parallel to the direction of the force; B does NO WORK when a particle is displaced because the direction of the motion is perpendicular to the direction of the force. –Work is a dot product of force and displacement; Fdx = Fxcos ; cos 90 = 0 –Since no work is done on the moving charged particle, the kinetic energy of the charged particle cannot be changed by a magnetic field alone.

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Differences Between E and B When a charged particle with a velocity v moves in a magnetic field, the magnetic field can change the direction of the velocity vector, but in cannot change the speed of the charged particle. Unit of the magnetic field: Tesla, T. –Tesla is a weber per square meter (Wb/m 2 ). –English unit is the Gauss (G); 1 T = 1 x 10 4 G.

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Example 29.1 Proton Moving in a Magnetic Field A proton moves with a speed of 8 x 10 6 m/s along the x-axis. It enters a region where there is a field of magnitude 2.5 T, directed at an angle of 60° to the x-axis and lying in the xy plane. Calculate the initial magnetic force and acceleration of the proton.

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Example 29.1 Proton Moving in a Magnetic Field If the particle had been an electron, do not use the negative sign of the charge in the calculation. We will continue to let the direction of the vector determine the sign of the vectors associated with magnetic fields. Use the right-hand rule to determine the direction of F mag : point the fingers of the right hand in the direction of v; the palm of the right hand is in the direction of the magnetic field B; and the thumb points up in the direction of F mag (the positive z direction). If you would rather work with vectors at right angles to each other, you can resolve the velocity vector into a component that is parallel to B and a component that is perpendicular to B.

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Example 29.1 Proton Moving in a Magnetic Field The velocity component parallel to B will have an F mag = 0 N. The velocity component perpendicular to B will have an F mag = q·v·B·sin 90. Acceleration:

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Magnetic Field Lines Magnetic fields can be represented with field lines just as we used for electric fields. The direction of the tangent ot a magnetic field line at any point give the direction of B at that point. The spacing of the lines represents the magnitude of B, the magnetic field is stronger where the lines are closer together.

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Magnetic Field Lines For the bar magnet: the magnetic field lines all pass through the magnet and form closed loops. The external magnetic effects are strongest near the ends where the field lines are most closely spaced. Because the magnetic field has direction, the field lines enter one end of a magnet and exit the other end.

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Magnetic Field Lines The end of a magnet from which the field lines emerge is called the North pole; the end of the magnet where the field lines enter the magnet is the South pole. Magnetic field lines always begin on the North pole and end on the South pole of a magnet. Opposite magnetic poles attract each other; like magnetic poles repel each other.

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Magnetic Field Lines

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Magnetic field lines that emerge from a plane are illustrated by a dot (the arrowhead) Magnetic field lines that go into a plane are illustrated by a cross (the tail of the arrow).

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Magnetic Field Lines

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Earth’s Magnetic Field The south pole of Earth’s magnetic field is the direction the north pole of a compass points towards, so we call it the north pole. –The magnetic field lines of Earth generally point down into Earth. The north pole of Earth’s magnetic field is the direction the south pole of a compass points towards, so we call it the south pole. –The magnetic field lines of Earth generally point away from the Earth.

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