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Transcript
Lecture 8 : Magnetic Forces & Fields • Magnetic fields • Biot-Savart Law • Ampere’s Law Recap • We have so far focussed on how charges give rise to electric fields, which exert forces on other charges • Moving charge constitutes a current, and creates new phenomena not described by electric forces : magnetism • We will now begin to explore deep connections between these different phenomena Magnetic Fields • Magnetism makes us think of a “classical magnet” with a north and south pole • However, more fundamentally it is a phenomenon produced by electrical currents Magnetic Fields • There are forces between current-carrying wires – even though those wires are electrically neutral! 𝐼2 𝐼1 𝐹 𝐹 𝐼1 𝐹 𝐹 𝐼2 • These are known as magnetic forces Magnetic Fields • We say that a current (or magnet) sets up a magnetic field in the space around it. This field causes another current (or magnet) to feel a force. Magnetic Fields 𝐼1 𝐹 𝐹 +𝑄 +𝑄 𝐹 𝐹 𝐼2 • Charges set up an electric field 𝐸 in the space around them • Currents set up a magnetic field 𝐵 in the space around them • An electric field 𝑬 is a region of space in which a charge experiences a force • A magnetic field 𝑩 is a region of space in which a current experiences a force Magnetic Fields How do we determine the strength of the magnetic field from a given current? Units: magnetic field strength is measured in Tesla (T) Biot-Savart Law Coulomb’s Law for 𝐸 𝑑𝐸 Biot-Savart Law for 𝐵 𝑑𝐵 𝑃 𝑃 𝑄 𝑑𝐸 = 𝑟 2 4𝜋𝜀0 𝑟 𝑟 𝑟 𝜇0 𝐼 𝑑 𝑙 × 𝑟 𝑑𝐵 = 4𝜋 𝑟 2 𝐼 +𝑄 𝑑𝑙 • 𝐸 is parallel to 𝑟 • 𝐵 is perpendicular to both 𝑑 𝑙 and 𝑟 • The force strength depends on the permittivity of free space 𝜀0 where 1 = 9 × 109 • The force strength depends on the permeability of free space 𝜇0 = 4𝜋 × 10−7 4𝜋𝜀0 Biot-Savart Law • Uses the concept of a cross product of two vectors, which is a vector perpendicular to each • The mathematical definition: 𝑐 𝑏 𝑎 Biot-Savart Law • What is the 𝐵-field around a wire? 𝜇0 𝐼𝑑 𝑙 × 𝑟 𝑑𝐵 = 4𝜋 𝑟 2 Direction can be recalled using the right-hand rule … Field traces out concentric circles … 𝐼 Thumb points in direction of current flow Fingers curl in direction of magnetic field 𝐵 𝑟 𝐵 Biot-Savart Law • What is the 𝐵-field around a wire? 𝜇0 𝐼𝑑 𝑙 × 𝑟 𝑑𝐵 = 4𝜋 𝑟 2 • We apply the Biot-Savart Law by splitting the wire into chunks 𝑑 𝑙 • 𝑑𝐵 from each chunk is parallel, so we can sum up the magnitudes 𝑑𝐵 𝜇 𝐼𝑑𝑙 sin 𝜃 • Biot-Savart : 𝑑𝐵 = 0 from this 4𝜋 𝑟2 chunk, where 𝑑 = 𝑟 sin 𝜃 and 𝑙 = 𝑟 cos 𝜃 𝑑 𝐼𝑑𝑙 𝜃 𝑟 𝐵 • Substituting in for 𝑟 and 𝑙 and integrating over 𝜃 from 0° to 180°, we find 𝜇0 𝐼 𝐵= 2𝜋𝑑 Ampere’s Law • Because of its vector nature, the Biot-Savart law is complicated to apply! Luckily there is a simpler formulation (analogous to Gauss’s Law) • Ampere’s Law states that the line integral of the magnetic field 𝐵 around a closed loop is equal to the current enclosed multiplied by 𝜇0 • In mathematical terms : 𝐵. 𝑑 𝑙 = 𝜇0 𝐼𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 Ampere’s Law What is meant by a line integral 𝐵. 𝑑 𝑙 of a vector field 𝐵? • Along a curve joining two points, sum up the elements 𝐵. 𝑑 𝑙 = 𝐵 𝑑𝑙 cos 𝜃 𝑑𝑙 𝐵 𝐵 • If the line forms a closed loop (i.e. has no beginning or end) then we write the integral as 𝐵. 𝑑𝑙 Ampere’s Law Why does Ampere’s law 𝐵. 𝑑𝑙 = 𝜇0 𝐼 work? 𝐵 • Consider a current 𝐼 which sets up 𝜇 𝐼 magnetic field 𝐵 = 0 2𝜋𝑟 ∆𝜃 • Draw any closed loop around 𝐼 𝐼 • Consider the bit of the loop passing between two radii separated by angle 𝑑𝜃 • The contribution 𝐵. 𝑑 𝑙 = • Total 𝐵. 𝑑𝑙 = 𝜇0 𝐼 2𝜋 𝜇𝑜 𝐼 .𝑟 2𝜋𝑟 𝑑𝜃 = 𝜇0 𝐼 𝑑𝜃 Ampere’s Law • What is the 𝐵-field around a wire? 𝐵. 𝑑 𝑙 = 𝜇0 𝐼𝑒𝑛𝑐 • We apply Ampere’s Law around a circular closed loop 𝐿 of radius 𝑑 • By symmetry, 𝐵 has the same strength around 𝐿 𝐼 • 𝐵 and 𝑑 𝑙 are parallel around 𝐿, such that 𝐵. 𝑑 𝑙 = 𝐵 × 2𝜋𝑑 𝑑 𝐿 𝐵 • The current enclosed by 𝐿 is 𝐼 • Ampere’s Law: 𝐵 × 2𝜋𝑑 = 𝜇0 𝐼 𝜇0 𝐼 𝐵= 2𝜋𝑑 Ampere’s Law Electric fields Magnetic fields • The electric field around a charge is given by Coulomb’s 𝑄 Law 𝑑𝐸 = 2𝑟 • The magnetic field around a current is given by the Biot- • The equivalent Gauss’s Law 𝑄𝑒𝑛𝑐 𝐸. 𝑑𝐴 = is easier to • The equivalent Ampere’s Law 𝐵. 𝑑 𝑙 = 𝜇0 𝐼𝑒𝑛𝑐 is easier to apply in practice 4𝜋𝜀0 𝑟 𝜀0 apply in practice • This integral is over a closed surface enclosing charge 𝑄𝑒𝑛𝑐 Savart Law 𝑑𝐵 = 𝜇0 𝐼 𝑑 𝑙 × 𝑟 4𝜋 𝑟 2 • The integral is around a closed loop enclosing current 𝐼𝑒𝑛𝑐 Summary • An electric current 𝐼 (or magnet) sets up a magnetic field 𝐵 around it; this causes another current (or magnet) to feel a force • The magnetic field 𝑑𝐵 due to a current element 𝐼𝑑𝑙 is given by the Biot-Savart Law 𝑑𝐵 = 𝜇0 𝐼𝑑 𝑙×𝑟 4𝜋 𝑟 2 • A more convenient formulation is Ampere’s Law 𝐵. 𝑑𝑙 = 𝜇0 𝐼𝑒𝑛𝑐 evaluated around a closed loop