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Vector Multiplication

Vector Multiplication. There are two ways (in 2 or 3D) to multiply vectors. Scalar product -> two vectors make a scalar. A ● B = N. Also called the dot product or the inner product. Vector product -> two vectors make a vector. Also called the cross product, alternating product or the

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Vector Multiplication

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  1. Vector Multiplication There are two ways (in 2 or 3D) to multiply vectors. Scalar product -> two vectors make a scalar A●B =N Also called the dot product or the inner product Vector product -> two vectors make a vector Also called the cross product, alternating product or the outer product A x B = C

  2. Scalar Product Scalar product -> two vectors make a scalar A ●B =ABcos q Geometric A ●B =axbx+ayby+azbz Algebraic

  3. Vector Product Vector product -> two vectors make a vector A x B = C Geometric C has magnitude absinq. Direction perpendicular to the plane containing A and B. A x B = (aybz-byaz)i+(azbx-bzax)j+(axby-bxay)k Algebraic

  4. The right hand rule Force F velocity v Magnetic Field B F=q(v x B)

  5. Electricity and Magnetism • One of the four fundamental forces of nature • Responsible for the vast majority of what we observe around us • Probably best-understood and best-tested of the forces of nature • Electromagnetic Interactions: • Electricity and Electronics • Magnetism • Chemistry • Biology • and even more

  6. Electrical Charges • Electric forces only affect objects with charge • Charge is measured in Coulombs (C). A Coulomb is a large unit of charge. 1 electron has -1.6 x 10-19 C of charge. • Charge comes in both positive and negative quantities • Charge is conserved – it can neither be created nor destroyed • Charge is usually denoted by the letter q. • An object has a total charge of 5 mC. It is divided into two pieces, one of which has charge 8 mC and the other of which has charge • 3 mC • -3 mC • 13 mC • Such a division is impossible

  7. Matter and Charges • All matter is made of positive and negative charges (or neutral) • An object’s total charge is very close to zero • When an object becomes charged, a tiny fraction of its charged particles (usually electrons) are lost or gained • These particles (usually electrons) can flow through objects What do you think of when you hear the words conductor or insulator? Conductor A material that allows electrons or other charged particles to flow freely Insulator A material that resists the flow of electrons and other charged particles

  8. Elementary Charge • Charges exist in integer multiples of a fundamental charge unit called e • We will consider e to be a positive number (some sources treat it as negative) e = 1.602 × 10-19 C the magnitude of the charge on an electron. the charge on a proton When you write that an atom or molecule has a charge +1 , you mean +e. A partial charge means that a charge density can be modeled as having a charge at a location less than e

  9. A cube with side 1 cm has a charge density of  = 1 C/m3. What is the charge of the cube? • 1 C • 0.01 C = 10 mC • 10-4 C = 100 m C • 10-6C = 1 mC 1 cm Charge Densities Charge can be localized to discrete points (point charges), or it may be spread out over a volume, a surface or a line • Charge density  units C/m3 • Surface charge density  units C/m2 • Linear charge density  units C/m

  10. Coulomb’s Law • Like charges repel, unlike charges attract • Force is directly along a line joining the two charges q1 q2 r 0= 8.85410-12 C2/ (N●m2) • Permittivity of free space • An inverse square law, just like gravity • Can be attractive or repulsive – unlike gravity • Constant is enormous compared to gravity

  11. ke= 8.988109 Nm2/C2 Coulomb’s Law: Applied A Helium nucleus (charge +2e) is separated from one of its electrons (charge –e) by about 3.00 10-11m. What is the force the nucleus exerts on the electron? Is it attractive or repulsive? r = 3.00 10-11m • We just calculated the force on the electron from the nucleus. How does this compare with the force on the nucleus from the electron? • The force on the nucleus is twice as big • The force on the nucleus is half as big • The forces are equal in magnitude q1 = 3.204 10-19C q2 = -1.602 10-19C • How does the acceleration of the nucleus compare to that of the electron? • The acceleration of the nucleus is larger • The acceleration of the nucleus is smaller • The accelerations are equal. Fe= - 0.513 N Attractive Force

  12. Newton’s Laws and Kinematics Newton’s laws and all the kinematics you learned in 113 are still true! If a does not depend on time, then

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