Does a change in electron's direction affect direction of force?

Question

A current carrying conducting wire is placed in a magnetic field, and hence, accelerates. Assuming the wire is moving in a constant magnetic field, will the electrons change direction due to the field?

If so, would that change the direction of the Lorentz force acting on the conductor?

Will the electrons ''bending'' in such a wire alter the direction of the force acting on the wire?

Answer 1

How are the two forces experienced in practical circumstances?

• in a motor, the induced electric field is called back emf and it opposes the voltage applied at the terminals of the motor.
• in a generator, the induced mechanical force is called amateur reaction or torque reaction and it tends to make the generator slow down or oppose whatever machine is been used to turn the generator.

In an ideal motor or generator (100% efficiency), the change in force would only be directional i.e.,

• from electric (I_y) to mechanical (F_y) or
• from mechanical (v_x) to electric (E_y).

Oops now I feel like I have written a text book!

Answer 2

Your question is a bit vague because it does not state whether the movement of the wire (and its electrons) is restricted to a particular direction.

When free electrons move in a magnetic field, they adopt a circular path given by the left-hand rule (Lorentz Force). Since only the component of motion that is perpendicular to the field will be converted into circular motion, the electrons could move in a helical path (shaped like a spring).

(Actually, much of the time the electrons would move in expanding or contracting spiral paths because circular paths can only be attained within a specially engineered environment such as a cyclotron or klystron tube. These are used to generate microwaves).

If the wire is free, it would also move in a circle because the free electrons in it are also trying to move in circles. The idea of a free wire is vague because current carrying wires have to be

• shaped as a circu-lar loop to complete the circu-lar circu-it and
• supported mechanically from some kind of pivot.

The loop shape and mechanical support significantly alter the direction of the net forces acting on the wire. Some parts of the loop may be outside the magnetic field or may interact with the same magnetic field at different angles.

In most theoretical cases and in machines such as motors:

• the electrons are restricted to move in only one direction which is along the length of the wire,
• the wire is restricted to move in only one direction perpendicular to the direction of current and perpendicular to the magnetic field,
• only one small straight section of the wire (current element) is considered for theoretical analysis,
• the analysis is only considered for short displacement where the motion is assumed to be along a straight line.

I suggest you play around with a real magnet, and real current carrying wires to see exactly how the shape, length, restrictions on a wire affect the direction of force.

---

Thanks for clarifying the question. I will assume a short piece of straight wire restricted to move perpendicular to the magnetic field.

Assume that

• the magnetic field is in the z-direction as B_x,
• the length of wire is in the y-direction as L_y,
• the wire is moving in the x-direction at a velocity v_x.

The electrons in the wire will have two perpendicular components of motion;

• in the y-direction along the length of wire due to the emf / voltage applied or current I_y
• in the x direction due to the motion of the wire at a velocity v_x

Consequently, there will be two forces:-

• MECHANICAL force F_x due to a magnetic field B_z acting on a CURRENT carrying wire (due to the motor effect given by the Left-Hand-Rule;
• ELECTROMOTIVE force (emf) F_y due to a magnetic field B_z acting on a MOVING wire (due to the generator effect (Faraday's Induction Law) given by the Right-Hand-Rule.

These TWO forces can are expressed mathematically as follows (note the vector / direction notation);

• F_x = B_z . I_y . L_y
• -F_y = B_z . v_y . Q_y

Note that the term electromotive force refers to potential difference or voltage (energy per charge) and not force in the acceleration (Newtons) sense. The term emf can only be expressed as force if the charge Q_y along the wire is known. Since we don't know how much charge there is in the wire (or how much of it is contributing to the current), the emf is better left in potential difference or voltage V_emf-y form as follows;

• -V_emf-y = B_z . v_y . L_y

Or as an electric field E_y as follows:

• -E_y = B_z . v_y

(Note the minus sign is due to the conventional positive current and Lenz's Law)

Note also that the electrons do not care whether the motion is electrical or mechanical; all they care about is moving in a circular path according to Lorentz's Law. Both the MECHANICAL force and the ELECTROMOTIVE force are exactly the same manifestation of Lorentz's Law viewed from different perspectives i.e.,

• one is Electric (current) -> Magnetic (field) -> Motion (i.e., how motors operate)
• the other is Motion -> Magnetic -> Electric (field) (i.e., how generators operate)

Back to your question

"...I'm concerned about that change in direction being a reduction of my calculated force..."

Your statement is essentially correct, except that it implies a simplistic case of scalar reduction. The motion of the wire due to the first force F_x will cause a change in force and direction by causing (actually inducing) a second force F_y that is perpendicular to the first.

The two forces are MUTUALLY PERPENDICULAR and should be expressed as vectors and not as a simple case of addition or subtraction.

The mutually perpendicular nature is what keeps electrons moving in circles according to Lorentz's Law (see how mass spectroscopy, cyclotrons and CRTs work).

Answer 3

The external force on the current carrying charges in the wire is approximately given by

$$ \int \rho\mathbf E_{ext} + \mathbf j\times \mathbf B_{ext}\,d^3\mathbf x. $$ where the $\mathbf E_{ext}, \mathbf B_{ext}$ are external fields.

The current density $$ \mathbf j = \rho \mathbf v, $$

where $\rho$ is charge density and $\mathbf v$ is their velocity. When the wire moves, the velocity of the wire adds to the velocity of the charges with respect to the wire, so $\mathbf j$ changes. The external magnetic force thus changes too, so that is always perpendicular to $\mathbf v$.