# 4.1 Unit 4: Magnetic Fields

Peter Apps

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## 4.0: Overview 👀

In previous units, we looked at how the electric field allows charged objects to interact without contact. In Unit 4, we'll take a look at magnetic fields, how they are created, and how they interact with electric fields. We'll cover how magnetic fields impact motion and interact with other magnetic fields.
Magnetic fields makeup 17-23% of the AP exam and will take between 13 and 26 class periods to cover, depending on the length of the class period. In AP Classroom, you can check out the Unit 4 Personal Progress Check that has ~30 multiple choice questions and ~1 free response question for you to practice on.

## 4.1: Forces on Moving Charges in a Magnetic Field 🧭

### What Does a Magnetic Field Look Like?

You may remember playing with magnets in elementary school, using a compass or iron filings to show the shape of the field. Magnetic field lines show the direction that the north pole of a magnet will be pushed or pulled. Just like electric charges, we can use the rule "likes repel, opposites attract". So, a north pole will repel other north poles and be attracted to south poles. Also, like electric fields, we can tell the strength of the field visually by looking at how close the lines are together.

Image from stickmanphysics.com

The Earth also produces a magnetic field that protects us from a variety of cosmic radiation. The charged particles from the solar wind spiral along the magnetic field lines and collect around the poles resulting in auroras. In order for this to occur, particles need to be charged to be affected by the Earth's magnetic field.

Image from wikipedia.com

Image from hyperphysics.phy-astr.gsu.edu/

Magnetic field strength is represented by B and has units of Tesla (T) where 1T = {Ns}/{Cm}. (Aren't you happy we can just write Tesla instead of Newton second per Coulomb meter? I am! 😄)

### Magnetic Force & the Right Hand Rule for Magnetic Fields 🖐️

So, why do charged particles curve in a magnetic field? They experience a force! The force from a magnetic field can be calculated using this equation:
Looking at this equation, we can see that in order to be affected by a magnetic force:
1. The object must be charged q =/= 0
2. The particle must be moving v =/= 0
3. There must be a magnetic field B =/= 0
4. The particle and the field must have a perpendicular component (cross product)
Ok, so we can calculate the magnitude of the force, but what about its direction? This can be easily found using the Right-Hand Rule (RHR). Point your thumb in the direction a positive charge is moving, your other fingers in the direction of the magnetic field, and your palm will face the direction of the force.

Image from schoolbag.info

Note that the RHR tells you information for a positive charge. If you have an electron (or other negative particles) you can still use the RHR, but the force will be facing in the opposite direction (into your palm instead of out of it). You can also use your LEFT HAND when you have a negatively charged particle to do the right hand rule and follow the same rules as the RHR.
Based on the way that the magnetic field is pointing or the way that charge is moving, you may use a Right-Hand Rule that your teacher taught you that looks more like a thumbs up 👍 or like an xyz plane with your middle finger pointing up in the air (😳) while your thumb points to the right and your pointer finger goes forward. The image above though is the best and most commonly used RHR, but the others are still valid choices based on the movement of charges!

### Paths of Charged Objects 🗺️

Because the magnetic force will always be perpendicular to the velocity of the moving charge, the field will cause the particle to curve as it moves. While there are many cases where the motion is not uniform circular motion (UCM), a majority of the questions involving calculations will assume that it is UCM. (For a review of this, check out this Fiveable Study Guide)

Image from courses.lumenlearning.com/

This concept is incredibly useful for detecting and directing moving charged particles. It's used in everything from old-style TVs to particle accelerators. A cathode-ray tube inside an old tv used magnetic fields to steer a beam of electrons to an exact spot on the screen where they would hit a fluorescent material and produce color.

Image from wikipedia.org

Physicists also use magnetic fields to help identify the particles created after a collision inside a particle accelerator. In the image below the green CCW spiral is an electron, and the purple CW spiral is its antiparticle, a positron.

Image from hst-archive.web.cern.ch/

### Magnetic & Electric Field Interactions 🏗️

There are also useful devices that use both electric and magnetic fields to exert two different forces on the charged particle. For example, the cathode ray tube in an old TV would use magnetic fields to steer the electrons vertically and electric fields to steer the beam horizontally allowing for an exact x,y position to be targeted.
These fields can also be aligned so that the two forces are opposing each other, allowing the particles to travel in a straight line. Devices, such as a mass spectrometer, use varying electric fields and magnetic fields to allow only particles with a certain velocity to pass through without deflecting.
The particles then enter a region with only a magnetic field, causing them to undergo UCM. The radius of the path allows scientists to determine the mass of the particle and understand more about the chemical makeup of the source material.

Image from openstax.org

## Practice Questions

1)
Two plates are set up with a potential difference V between them. A small sphere of mass m and charge -e is placed at the left-hand plate, which has a negative charge, and is allowed to accelerate across the space between the plates and pass thourgh a small opening. After passing through the small opening, the sphere enters a region in which there is a uniform magnetic field of magnitude B directed into the page, as shown above. Ignore gravitational effects. Express all algebraic answers in terms of V, m, e, B, and fundamental constants, as appropriate.
(a) i. What is the initial direction of the force on the sphere as it enters the magnetic field? (Check one.)
_ Into the page, _ Out of the page, _ Towards the top of the page, _ Towards the bottom of the page
ii. Describe the path taken by the sphere after it enters the magnetic field.
(b) Derive an expression for the speed of the sphere as it passes through the small opening.
(c) Derive an expression for the radius of the path taken by the sphere as it moves through the magnetic field.

a) i. Towards the bottom of the page. Use the RHR, then remember that the RHR is for positive charged objects, so switch the direction of the force because we have an electron.
ii. With a net force pushing towards the bottom of the page, the particle will travel in a circular path curving towards the bottom of the page.
b)
c)
2)

Image from apclassroom.collegeboard.org

The figure above shows the paths of five particles as they pass through the region inside the box that contains a uniform magnetic field B directed out of the page. Which particle has a positive charge?

Use the RHR with your fingers pointing out of the page.
A - Force is directed upwards, but shows a downward curve. A must be negatively charged
B - Force is directed downwards, and the curve is downwards. B must be positively charged
C - Same as A, must be negatively charged
D - Force is downwards, curve is upwards. D must be negative as well
E - No curve, so no charge. E is neutral
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