# Electric lines of force

This is one of the questions that were asked in an exam:

Assuming electric lines of force are nothing but the electric field lines (correct me if I'm wrong):

Firstly, how can an electric line of force be given by $x^2 +y^2=1$. That's the equation of a circle with a radius of 1 unit. And since a circle is a closed loop, this would imply that the electric line of force forms a closed loop, which they do not.

Secondly, even if we restrict the curve to any two quadrants, say the $I^{st}$ and the $II^{nd}$ quadrants will it be correct to say that the force at the point $(1,0)$ will be along the tangent to that point? (That's what I thought but the answer key says that the particle will move along the circle).

• I assume they mean an equipotential, remember that $E = -\nabla V({\bf r})$. So in this case, the electric field is radial pointing towards the center – caverac Apr 15 '17 at 12:25
• The tangent is along the circle. – ZeroTheHero Apr 15 '17 at 13:13
• It could be an induced electric field ; that may be why it's circular. – Gauri Apr 15 '17 at 14:09

It is possible to have circular field lines when the electric field is induced from a time-varying magnetic field. The $\vec E$ field does not result from static electric charges, and is not conservative.

The image below, taken from a textbook by Halliday, Resnick and Krane, illustrates an example.

Basically the argument is that a variable magnetic flux through an open surce would induce a current as per $$-\frac{d}{dt}\int \vec B\cdot d\vec S=\oint \vec E\cdot d\vec \ell \ne 0\, .$$ In this case the electric field need not be curl-free and the electric field lines need not start of end at a charge.

• The image is from Fundamentals of Physics by Halliday, Resnick, and Walker (8th ed.). – Apoorv Potnis Dec 3 '17 at 16:20

No I do not think the circular electric line of force is an equipotential. Unlikely as the situation may seem, as far as answering the question is concerned you must accept it. There can be circular lines of force (induced electric fields) if a magnetic field is changing.

You are correct that the force on the particle at (1,0) will be along the tangent to the circle. This is more accurate than saying that the particle will move along the circle. The given answer (c) is ambiguous. If the particle has inertia it will not move along the line of force - ie around the circle - because it will accelerate (the force is tangential not radial) and inertia will carry it away from the circle.

The line of force is the same as the direction of the electric field. It only tells you the direction of the force on the particle at that point. This force will probably cause the particle to move away from this line of force, onto another line, where the force will be in a different direction. A line of force or electric field line is not the same as the line of motion (trajectory) of the particle.

• So the answer should be (b): will move along a straight line? – Kunal Pawar Apr 15 '17 at 14:54
• @KunalPawar You have all the information to figure it out by yourself. – ZeroTheHero Apr 15 '17 at 16:54
• @ZeroTheHero And hence I conclude that the answer is (b) -.- – Kunal Pawar Apr 15 '17 at 16:59
• @KunalPawar No I am saying that the question is a very ambiguous (badly worded). The initial (instantaneous) motion of the particle is along the tangent, but inertia will also carry it radially onto another line of force, where it gets a tangential kick, then onto another line of force, and so on. The long-term motion will be an outward spiral, if all lines of force are concentric circles. ... I do not think it is worth bothering too much about which answer is correct. Personally I would opt for (d). – sammy gerbil Apr 15 '17 at 17:15
• @sammygerbil Indeed it is ambiguous. Perhaps the best thing would be to not attempt it at all. – Kunal Pawar Apr 15 '17 at 17:19

The E-Field Vector is tangential to the lines of force. But for a particle to move in a circular motion the force needs to be pointing to the center of the circle. Apart from that just to have a single line of force is already a bit scetchy.

Ask a remark on the question: the question is problematic and indicates a lack of understanding for the subject.

• "If you can change school fast, dont waste your time there." ??????? – ZeroTheHero Apr 15 '17 at 16:52
• @ZeroTheHero Precisely what I thought. '???????' indeed. – Kunal Pawar Apr 15 '17 at 17:00
• @sammygerbil I take no offence though. For the exam is not one that was conducted by my school. It's one of the questions I found in an entrance examination for a college. – Kunal Pawar Apr 15 '17 at 17:21
• @sammygerbil : good point. I was a bit shocked by the question. maybe now it is clearer. – lalala Apr 15 '17 at 17:22