physics - clutch non-calc ch 28: sources of magnetic...

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PHYSICS - CLUTCH NON-CALC

CH 28: SOURCES OF MAGNETIC FIELD

CONCEPT: MAGNETIC FIELD PRODUCED BY MOVING CHARGES

Remember: A charge moving through an existing Magnetic Field FEELS a Magnetic FORCE.

EXAMPLE: A 3 C charge is moving right with a constant 4 m/s. What is the magnitude and the direction of the magnetic field that this charge produces 2 cm directly above itself?

ALSO: A moving charge __________________________________________ (much less popular question): - MAGNITUDE ______________________ - Remember µ𝐨 = 4π*10-7 N/A2 = 1.26*10-6 N/A2

- Angle Θ is between _____ and _____, which is a vector between charge and location of produced field - DIRECTION comes from RIGHT HAND RULE, by “grabbing” the LINE OF MOTION.



CONCEPT: MAGNETIC FIELD PRODUCED BY STRAIGHT CURRENTS

Remember: Moving Charges PRODUCE A NEW FIELD B = ___________________

EXAMPLE 1: What is the direction of the magnetic field produced by a current on a very long wire if the current is oriented:

(a) up (b) left (c) into the page EXAMPLE 2: The two wires shown below are 4 m away from the other. What is the magnitude and direction of the magnetic field that is produced at a point in the center of the two wires?

Currents are just charges moving in a wire. So currents ALSO produce new Magnetic Fields:

- MAGNITUDE ______________________ (for very long wire) - Remember µ𝐨 = 4π*10-7 N/A2 - DIRECTION RIGHT HAND RULE: GRAB wire, with THUMB in direction of ___________________. - TWO Fields at same location: Same Direction _______________ Opposite _______________

. i = 3 A i = 5 A



EXAMPLE: FIND MAGNETIC FIELD DUE TO TWO PERPENDICULAR CURRENTS

The very two long, perpendicular wires below intersect at (0, 0). The vertical wire carries 2 µA up, while the the horizontal

wire carries 3 µA to the left. What is the net magnetic field at point P located at (-4, -9) cm?

EXAMPLE: FIND ZERO MAGNETIC FIELD Two long, horizontal wires are 6 m away from each other. The bottom and top wires carry currents of 4 A and 5 A, respectively, both to the right. How far from the bottom wire is the NET magnetic field due to these currents zero?



CONCEPT: MUTUAL MAGNETIC FORCE ON PARALLEL CURRENTS

Remember: Current-carrying wires PRODUCE NEW Magnetic Fields B = ______________ - A current-carrying wire in an EXISTING Field FEELS A FORCE F = ______________

EXAMPLE: Two horizontal wires 10 m in length are parallel to each other, separated by 50 cm. The top wire has current 2 A

to the right, and the bottom wire has current 3 A to the left. What is the magnitude and direction of the force exerted on the:

(a) top wire?

(b) bottom wire?

So if you have two PARALLEL currents, you get a MUTUAL Force between them: - MAGNITUDE: - Force per unit length = - DIRECTION: Same Direction _______________ Opposite _______________



PRACTICE: FORCE PER UNIT LENGTH ON PARALLEL WIRES Two very long wires of unknown lengths are a parallel distance of 2 m from each other. If both wires have 3 A of current flowing through them in the same direction, what must the force per unit length on each wire be?

BONUS: Is the mutual force between the wires attractive or repulsive?



CONCEPT: MUTUAL MAGNETIC FORCE ON PARALLEL CHARGES

Remember: Parallel currents FEEL A MUTUAL FORCE F =

EXAMPLE: An electron is moving right with 1.0 x 108 m/s when a proton passes it moving left with 2.0 x 108 m/s.

(a) What is the magnetic force between them when they pass each other, if at that moment they are 3 µm apart?

(b) What is the electric force between them at that moment?

Currents are just charges moving in a wire. So parallel moving charges ALSO feel a mutual Magnetic force: - MAGNITUDE _______________ - Same Direction & Charge _______________

- Opposite Direction & Charge _______________

- All Others Combinations _______________



CONCEPT: MAGNETIC FIELD PRODUCED BY LOOPS AND SOLENOIDS

Remember: Current-carrying wires PRODUCE NEW B-Fields B = ______________ - In STRAIGHT wires: Current is STRAIGHT _______________ B is CURVED ______________ - In wire LOOPS: Current CURVES _______________ B is STRAIGHT ______________ (1) Single or Multiple Loops B = _______________

(2) Solenoid (very long loop) B = _______________

- Solenoids produce magnetic fields similar to ________________!

Remember: TWO Fields at same location: Same Direction ADD Opposite SUBTRACT

EXAMPLE: A wire is twisted into 5 tight loops 4 m in radius. A 3 A current is ran through the wire in the counter-clock direction. Find the magnitude and direction of the magnetic field produced by the loop in its center.



EXAMPLE: HOW MANY TURNS IN A SOLENOID How many turns should a 2 m solenoid have in order to produce a 0.4 T magnetic field when a 3 A current is ran through it? PRACTICE: FIND CURRENT IN LOOP PERPENDICULAR TO PAGE The single loop below has a radius of 10 cm and is perpendicular to the page (shown at a slight angle so you can better visualize it). If the magnetic field at the center is 10-6 T directed left, what is the magnitude of the current? What is the direction of the current at the top of the wire: into the page (towards left) or out of the page (towards right)?



EXAMPLE: DESIGNING A SOLENOID You are tasked with designing a solenoid that produces a magnetic field of 0.03 T at its center with a radius of 4 cm and length of 50 cm. What is the minimum total length of 12 A wire you should buy to construct this solenoid? PRACTICE: FIND CURRENT THROUGH SOLENOID A long wire having total resistance of 10 Ω is made into a solenoid with 20 turns of wire per centimeter. The wire is connected to a battery, which provides a current in order to produce a 0.04 T magnetic field through the center of the solenoid. What voltage must this battery have?



EXAMPLE: MAGNETIC FIELD BY TWO CONCENTRIC LOOPS Two wire loops are arranged concentrically, as shown below. The inner wire has diameter 4 m and clock-wise current 5 A. The outer wire has diameter 6 m and counter-clockwise current 7 A. What is the magnitude and direction of the net magnetic field that is produced at the center of the two loops?

PRACTICE: MAGNETIC FIELD BY TWO CONCENTRIC SOLENOIDS The two tightly wound solenoids below both have length 40 cm and current 5 A in the directions shown. The left solenoid has radius 20 cm and 50 m of total wire. The right solenoid has radius 0.5 cm and 314 m of total wire. The thinner solenoid is placed entirely inside the wider one so their central axes perfectly overlap. Assume wires don’t touch. What is the magnitude and direction of the magnetic field that is produced by a combination of the two solenoids at their central axis?



CONCEPT: MAGNETIC FIELD BY TOROIDAL SOLENOIDS Remember: Magnetic Field at the center of a LOOP B = - Magnetic Field at the center of a SOLENOID B = Solenoids can be arranged in a doughnut shape to form Toroidal Solenoids aka Toroids

B = _________________ - NOTE _____ is back, AND _____ NOT _____! - B exists between _____ and _____, zero outside. - R is “mean radius” = ___________________ EXAMPLE: A 300-turn toroidal solenoid has inner and outer radii 12 and 16 cm, respectively. If 5 A of current runs through the wire, what is the magnitude of the magnetic field produced:

(a) at the center of the solenoid

(b) at 14 cm away from the center

(c) at 20 cm away from the center



CONCEPT: BIOT-SAVART LAW WITH CALCULUS

Biot-Savart Law reduces to familiar equations:

- Point charge: 𝐵 =𝜇0

4𝜋

𝑞𝑣𝑠𝑖𝑛𝜃

𝑟2

- Current-carrying wire: 𝐵 =𝜇0𝐼

2𝜋𝑟

EXAMPLE: Show that the Biot-Savart law for a current is the same as the equation above for a point charge.

For ANY current, magnetic field 𝑟 away is

= __________________

- Known as Biot-Savart Law

i

𝑟



EXAMPLE: MAGNETIC FIELD DUE TO FINITE, CURRENT-CARRYING WIRE What is the magnetic field at the position shown in the following figure due to the finite, current-carrying wire?

i

z

x -a +a

z



PRACTICE: MAGNETIC FIELD AT CENTER OF RING OF CURRENT What is the magnetic field at the center of the following ring of current?

i

r



CONCEPT: AMPERE’S LAW WITH CALCULUS

- Like for Gauss’ law, the magnetic field depends ONLY on the current enclosed by an “Amperian loop”. EXAMPLE: Using Ampere’s law, find the magnetic field due to an infinitely long, current-carrying wire.

ANY magnetic field, , must satisfy the following equation:

∮ ⋅ 𝒅𝒍

𝑆= __________________

- Known as Ampere’s Law

S i

𝑑𝑙



EXAMPLE: MAGNETIC FIELD DUE TO A SOLENOID What is the magnetic field along the axis of a solenoid?



PRACTICE: MAGNETIC FIELD DUE TO SOLID, CYLINDRICAL CURRENT-CARRYING CONDUCTOR A solid, cylindrical conductor carries a uniform current density, 𝑱. If the radius of the cylindrical conductor is 𝑹, what is the

magnetic field at a distance 𝒓 from the center of the conductor when 𝒓 < 𝑹? What about when 𝒓 > 𝑹?

J



physics - clutch non-calc ch 28: sources of magnetic...

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