Magnetic Forces

Now that we understand how magnetic fields are generated let’s look at how they affect other charged particles.

Unlike electric fields which impact any charge that enter them, magnetic fields only act on moving charges. This means that while an electric field exerts a force on a stationary charge, a magnetic field would not. Since all forces are vector quantities we need to learn a second right-hand rule to predict a magnetic force’s direction in addition to a new set of equations to determine its magnitude.

The Second Right Hand Rule

The second right-hand rule picks up where the first one left off and allows us to predict the magnetic force. Instead of aligning our thumb with the current, we will point it in the direction of the moving charge. Since current is a collection of moving charges the two are analogous.

Again like the first right-hand rule the tips of our fingers point in the direction of the magnetic field. However, we will leave our fingers outstretched in the direction of the magnetic fields rather than curling them.

Lastly, our palm points in the direction of the magnetic field for a positive charge. I imagine my palm pushing the charge in whatever direction it is pointing to visualize the force more easily. Since negative charges are a bit backward the force exerted on them is represented by the back of our hands. I imagine backhanding a negative charge away to visualize the force in this scenario.

It pretty hard to visualize exactly how this works just from words so take a look and try and follow along with the different scenarios presented below.

Magnetic Magnitude

Now that we understand the direction of a magnetic force let’s learn how to determine a magnetic force’s magnitude. On the MCAT there are two different scenarios we need to know: a moving charge and a current-carrying wire. Since both equations are similar to one another I will present them together and walk through their similarities and differences.

Moving Charge:
\[ F_B=qvB\;sin\theta\]
Current-Carrying Wire:

Although the qv and IL in the above equations are different they represent the same information. Since q represents charge and v the velocity of that charge qv is the representation of the magnitude of the charge and its velocity. Since current tells us how much charge passes through a wire per second and L how long that wire is IL also represents the magnitude of charge in the wire and its velocity. Therefore both qv and IL stand for the same concept the magnitude of a charge and its velocity.

Here the three elements present in our second right-hand rule are represented in both equations: qv or IL = direction of charge movement, B = magnetic field, and FB = the magnetic force. Here the angle is the smallest angle between the velocity vector and the magnetic field vector.

Let’s look through a couple of sample questions to see how we can use this equation with the right-hand rule to determine a magnetic force vector.

Example: Mass Spectrometer

Mass spectrometers(MS) are frequently used to determine the identity of unknown compounds. MS devices accomplish this by fragmenting whole molecules and attaching positively charged ions to the fragments. These fragments are then subjected to a uniform magnetic field and the particle deflection within the field is measured. The amount of deflection depends on the mass to charge ratio of the particle (m/z) with heavier particles experiencing less deflection.

Moving Charge

If a fragment with a 3.2 x10-19 C charge and a velocity of 20 m/s passes through the 0.2T magnetic field of the mass spectrometer as shown below what force would it experience and in what direction?

Since the calculation is often the easiest part of these question I like to start by solving for the magnitude of the force first. That way I can eliminate any answers with the wrong magnitude and narrow down my answer choices giving myself a better chance of getting the question correct even if I don’t remember how to use the right-hand rule.

To start this question we will begin by finding the angle between the velocity vector and the magnetic field. Since we are looking at this top down it can be difficult to visualize what angle we are talking about. To make it easier I am going to rotate this picture so we are looking at it from the side.

In this scenario, the arrows point upwards rather than downwards because the circles represent arrows coming towards us and out of the page. From this view, the angle between the velocity vector and the magnetic field is 90°. Since the sine of 90° is 1 we can omit this part of the equation and plug in our q, v, and B value into the charge equation.

F_B= qvB \]
\[F_B=(3.2\times10^{-19}C) (20\frac{m}{s})(0.2T)\]
\[F_B= 12.8\times10^{-19}N\]

Now that we have found the magnitude of the force let’s use the right-hand rule to determine the direction of our force. To do this we will need to use the original picture that was given to us in the questions stem. Start by lining up your thumb with the velocity vector then point your fingers towards you by extending your wrist. The hand positioning is a bit awkward here but our palm should point up towards the ceiling showing us the direction of the magnetic field.

Therefore our positive charge will experience an upwards 12.8 x 10-19N magnetic force causing our particle to deflect in that direction. Lets walkthrough how to use the wire equation now so we can see how similar the setup is for both equations.

Current-Carrying Wire

If a wire with current of 3A and length of 1m was placed into the mass spectrometer’s 0.2T magnetic field as shown below what would the magnitude of the magnetic force on the wire be?

Again we will begin by finding the magnitude of the force on the wire. As with the previous questions, we will begin by finding the smallest angle between the current and the magnetic field or θ in our equation. From the picture above we can see that the smallest angle is 30° since sine 30° is 0.5 we can plug that value into our equation and solve using the remaining variables.

\[ F_B= ILB\;sin\theta\]
\[F_B= (3A)(1m)(0.2T)(0.5)\]
\[ F_B=0.3 N\]

Now that we have solved for the magnitude of the force let’s use our right hand rule to determine its direction. To begin align your thumb with the direction of the current then point your finger towards the ceiling in the direction of the magnetic field. Your palm should be facing you indicating the magnetic force will point out of the page at you.

Therefore the wire will experience a 0.3N magnetic force that points out of the page.

Conceptual and Passage Based

Now that we have seen how to use the calculation based questions let’s look at some more conceptual questions and apply what we have learned so far.

If a fragment with the same charge and velocity was analyzed in the same manner as in question 1 what would its deflection path look like?

For this question we already determined the direction of the magnetic force as shown below. Since the charge is already traveling to the left and the force acts upwards the particle will deflect upwards. This occurs because the force causes the particle to accelerate upwards and the longer the force acts on the charge the greater its velocity in that direction. Using our knowledge of vectors we can determine that the charge will move to the up and to the left.

If the fragment with the same mass, charge quantity, and velocity was negative instead of positive what would its deflection path look like?

In this question the only thing that has changed about charge is its sign. Previously we were dealing with a positive charge which deflected upwards in the magnetic field now that the sign has changed our charge will deflect in the opposite direction. While we could use the right hand rule for this question we don’t need to since positive and negative charges will behave in exactly the opposite fashion. This makes sense since our palm represents forces for positive charges while the back of our hand represents negative charges.

Now that we have walked through a variety of different question together try the quiz below to get additional practice with the right-hand rule, equations, and how the different concepts apply to a variety of scenarios.

Quiz Questions

If the direction of the magnetic field were reversed as shown below what path would the direction of the positive and negative charges?

Which of the following would particles would experience no magnetic force?

Which of the following atomic particles would experience the smallest acceleration in a magnetic field?

Which of the following charges would experience the greatest amount of deflection?