Whiteboard Warmup! A beam of electrons is being fired to the right, when a magnet is pointed toward the beam and brought closer. As a result, the beam of electrons curves upward, striking the top of the glass container. Which pole of the magnet was brought near? Show your reasoning with a diagram of the velocity, force and magnetic field that the electrons experience.

Inverse Right Hand (Left Hand) Rule The electrons arc upward, so the magnetic force is upward. Point the palm of your left hand upward. The velocity is to the right – turn your hand so that your thumb points to the right. Now your fingers should be pointing out of the page! Magnetic field comes out of the North end of a magnet, so this means that the North end of the magnet was pointed toward the electron beam.

In a B-field, the magnetic force exerted on a moving particle will always be perpendicular to its velocity vector. v F v F v F This means that magnetic force can never speed up a particle, and can never slow down a particle. It can only change the particle’s direction!

Magnetism, meet Mechanics. Mechanics, Magnetism. A proton of mass m and charge q is fired into the magnetic field B shown below at a velocity v. a)Show the complete path of the proton and show its velocity and the force and the exerted on it at multiple points. b)Derive an expression for the radius of the proton’s path! +q+q

Weeeeeee! Magnetic force causes the particle to travel along a circular arc. The radius of the arc can be derived by setting the magnetic force (qvB) equal to mv 2 /r. End result -> +q+q r

Whiteboard: Magnetic Loops B -q, m, v -q, m, 2v q, 2m, v 2q, 2m, v Four particles are fired into the magnetic field shown below. Sketch (to scale) the paths of each of the particles. Be sure to be consistent! Then, write an expression for the radius of the arc made by each.

In a B-field, the magnetic force exerted on a moving particle will always be perpendicular to its velocity vector. v F v F v F This means that magnetic force can never speed up a particle, and can never slow down a particle. It can only change the particle’s direction!

Another way of saying this is… Magnetic force cannot do work! A B-field can never add or remove kinetic energy from a system. It can only change the system’s direction of motion while maintaining a constant speed. Since kinetic energy is a scalar quantity, this will leave the system’s kinetic energy unchanged. W = FΔxcosθ Always 90° = 0 J

A magnetic field of 0.1 T forces a proton beam of 1.5 mA to move in a circle of radius 0.1 m. The plane of the circle is perpendicular to the magnetic field. 5)Of the following, which is the best estimate of the speed of a proton in the beam as it moves in the circle? (A) m/s (B) 10 3 m/s (C) 10 6 m/s (D) 10 8 m/s (E) m/s 6)Of the following, which is the best estimate of the work done by the magnetic field on the protons during one complete orbit of the circle? (A) 0 J (B) J (C) J (D) 10 2 J (E) J

A magnetic field of 0.1 T forces a proton beam of 1.5 mA to move in a circle of radius 0.1 m. The plane of the circle is perpendicular to the magnetic field. Of the following, which is the best estimate of the work done by the magnetic field on the protons during one complete orbit of the circle? (A) 0 J (B) J (C) J (D) 10 2 J (E) J

+q+q Suppose you wanted to hook up the plates to opposite terminals of a battery, so that the proton travels straight through the plates, undeflected. Which way would you need to hook up the battery?

+q+q Combining electric and magnetic forces! v E B If the strengths of the fields are fine-tuned so that the particle travels straight through, derive an expression for the velocity of the particle. F B = qvBF E = Eq

+q+q v E B FEFE FBFB The forces will only be balanced if qvB = Eq. Therefore, only particles with the exact velocity v = E/B will make it through. This device is called a velocity selector and is used in particle accelerators to hand-pick the right particles for a collision!

+q+q v E B What will happen if the particle is not moving fast enough? (v < E/B) ? What about if the particle is moving too fast? (v > E/B) ?

Too slow: Magnetic force too weak. Electric force dominates. Too fast: Magnetic force too strong. Magnetic force dominates. Juuuust right! F B = qvBF E = Eq

Beam of protons with randomly distributed speeds Whiteboard Challenge: Capture the Protons! m p = 1.67 x kg e = 1.6 x C B = 0.2 T E = 400 kV/m a) Sketch the complete path of the protons that will make it through the velocity selector undeflected. b) Where should a detector be placed along the orange wall (quantitatively) to measure the number of protons per second that made it through the velocity selector?

The only protons that make it through the crossed E and B fields must have a speed v = E/B. v = (400,000 V/m)/(0.2 T) = 2 x 10 6 m/s Once they are in the region of only B-field, they will immediately move in uniform circular motion with a radius given by r = 10 cm And you can use RHR #2 to determine which way they will curve!

Too slow: Magnetic force too weak. Electric force dominates. Too fast: Magnetic force too strong. Magnetic force dominates. 2r = 20 cm

Magnetic Flux The amount of magnetic field that is passing through a given surface. Magnetic flux depends on several factors. Let’s take a look! Φ

Magnetic flux is the amount of magnetic field that is passing through a given surface. Therefore, the stronger the field, the greater the magnetic flux through the loop! Small fluxLarger flux

Magnetic flux is the amount of magnetic field that is passing through a given surface. Therefore, the larger the area, the greater the magnetic flux through the loop! Small fluxLarger flux

Lastly, the more perpendicular the loop is to the field, the greater the magnetic flux through the loop! Max fluxLess fluxZero flux The more the magnetic field goes through the loop, the greater the flux will be.

How can we measure the relative direction of a field and a surface? Answer: With the area vector! A A The area vector points perpendicularly away from the surface, and is used to compare the orientation of the surface to the direction of an external magnetic field.

A B B A If the angle between the area vector and the B field is 0°, then there is a maximum magnetic flux through the loop. If the angle between the area vector and the B field is 90°, then there is zero magnetic flux through the loop.

A θ B Magnetic Flux B: Strength of magnetic field A: Area of loop θ: Angle between B and A Units: Tm 2 Derived unit: Weber (Wb)

Rank the loops, in order of increasing magnetic flux a)b)c)d) e)f)g)h)

ZeroBA/4BA/2BA2BA g = efb = da = hc