Question Video: Determining the Potential of a Conducting Rod in a Uniform Magnetic Field | Nagwa Question Video: Determining the Potential of a Conducting Rod in a Uniform Magnetic Field | Nagwa

Question Video: Determining the Potential of a Conducting Rod in a Uniform Magnetic Field Physics • Third Year of Secondary School

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A conducting rod moves through a uniform magnetic field, as shown in the diagram. Which end of the rod has a higher potential?

03:12

Video Transcript

A conducting rod moves through a uniform magnetic field, as shown in the diagram. Which end of the rod has a higher potential?

To begin, we should recall that the electric potential at a point helps tell us the amount of work it would take to move a test point charge, that is, a single positively charged particle, to that point. Generally, electric potential is greater near a positive charge than it is near a negative charge, since it takes more work to push a positive test charge closer to a positive charge than it would to push a positive test charge closer to a negative charge.

So, what does this have to do with a rod moving through a magnetic field? Well, we should also recall that when a straight conductor moves through a uniform magnetic field, there is an electromotive force, or emf, induced across it with a magnitude of 𝑙𝑣𝐡 sin πœƒ, where 𝑙 is the length of the conductor, 𝑣 is its speed, 𝐡 is the strength of the magnetic field, and πœƒ is the angle between the conductor’s velocity and the magnetic field.

In this question, we have a rod that’s moving to the right through a magnetic field that points into the screen. So the rod’s direction of motion is perpendicular to the magnetic field, and πœƒ equals 90 degrees. We know that the sin of 90 degrees just equals one. So here, the formula for the induced emf simplifies to 𝑙𝑣𝐡. Now, the length of the rod, its speed, and the strength of the magnetic field are all nonzero. So we know that there will indeed be an induced emf across this rod.

Thus, as the rod moves through the field, free charges in the rod will be accelerated toward the rod’s top or bottom. This buildup of negative charge at one end and positive charge at the other end constitutes a potential difference across the rod. So, if we can just determine the sign of the charge at either end, we can compare the electric potential at either end and answer this question. To do this, let’s first determine the magnetic force that acts on free electrons in the rod using the right-hand rule.

To determine the force on a particle in the rod, we need to first identify the direction of π‘žπ‘£, where π‘ž is the charge of the particle and 𝑣 is its velocity. We know that 𝑣 alone points to the right. But since we’re presently thinking about the force on an electron, this π‘ž-term introduces a negative sign to π‘žπ‘£. And so π‘žπ‘£ points left. Now, using our right hand, we point our fingers in the direction of π‘žπ‘£ and then curl them in the direction of the magnetic field. Here, the magnetic field points into the screen. So we have to flip our hand upside down in order to curl our fingers that way. Then, the thumb points in the direction of the force on the particle.

Since our right hand has to make a thumbs-down shape here, we know that free electrons in the wire experience a magnetic force that accelerates them downward. This means that as the rod moves, negative charge will build up at the bottom end of the rod, B. And likewise relative positive charge will build up at the top end of the rod, A.

Finally, we just need to compare the electric potential at the two ends of the rod. Since electric potential is generally greater near positive charge than it is near negative charge, we know that the end labeled A has a higher potential. So, when asked whether end A or B of this rod has a higher potential, we know that the answer is A.

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