A positive charge moves within regions containing magnetic fields, moving perpendicular to those fields. The moving charge is subject to magnetic forces, as shown in cases a, b, and c. In what direction is the magnetic field aligned in case a? In what direction is the magnetic field aligned in case b? In what direction is the magnetic field aligned in case c?
We see, off to the right, three diagrams: a, b, and c. And in each case, we wanna solve for the direction of the magnetic field which is not shown in the diagram. We can call these magnetic fields 𝐵 sub 𝑎, 𝐵 sub 𝑏, and 𝐵 sub 𝑐. And we’ll remember that we’re solving for direction only, not magnitude.
To help figure out these directions, let’s recall the mathematical relationship for the magnetic force on a charged particle. The magnetic force 𝑭 sub 𝐵 on a discrete charge 𝑞 is equal to that charge times the cross product of its velocity 𝒗 in the magnetic field that it exists in, 𝐵. The cross product in this equation shows how we can think about directions in order to solve for the directions of the magnetic fields in our three diagrams. Specifically, we can find these directions out using a rule that involves our right hand. Sometimes called a right-hand rule.
In this arrangement of our right hand, we have our thumb, our index finger, and our middle finger pointing in directions perpendicular to one another. If we point our thumb in the direction of the velocity 𝒗 and our index finger in the direction of the magnetic field 𝐵, then our middle finger, as a result, points in the direction of the magnetic force on the charge involved. We can use this right-hand rule with the diagrams shown — a, b, and c, — to solve for the magnetic field directions in each case.
Let’s start with the magnetic field in diagram a. If we point our thumb upward in the direction of the velocity 𝒗; stretch out our fingers as shown in the right-hand rule; and point our middle finger in the direction of the magnetic force 𝑭, and then we find our index finger, which according to our right-hand rule points in the direction of the magnetic field, pointing into the page. So 𝐵 𝑎, the magnetic field in diagram a points into the page.
Next, we move on to solving for 𝐵 sub 𝑏, the magnetic field in diagram b. We use the same approach pointing the thumb of our right hand down in the direction of the velocity 𝒗 and the middle finger of our right hand into the page, the direction of the magnetic force on this charged particle. When we do this using our right hand, our index finger, which indicates the direction of the magnetic field, points to the left. So that’s the way that 𝐵 sub 𝑏 points. In our diagram, it points to the left.
Finally, we move to the magnetic field 𝐵 sub 𝑐 in diagram c. When we point our thumb to the left in the direction of 𝒗 and our middle finger up in the direction of the magnetic force, our index or pointer finger points out at us out of the page. So that’s the direction of the magnetic field in diagram c. It points out of the page.