Electrostatic Fields, Electric Potential, and Electromotive Force: What Actually Drives Electric Charge?

A charge can move around a circuit loop after loop — where does the energy come from? If we consider only electrostatic force, the total work it does along a closed loop is zero, so it cannot by itself supply energy to the charge on every lap. To answer this question, we must first separate three levels: the electric field describes electric effects in space, force describes what a specific charge experiences, and electromotive force describes how much energy each unit of charge gains.

Why can an electrostatic field be described by electric potential?

Place a sufficiently small test charge$q$at some point in space. If the electric field force it experiences is$\mathbf F$, then we define

$\mathbf E=\frac{\mathbf F}{q}$

By convention, the direction of the electric field is the direction of the force on a positive test charge. If the charge distribution producing the electric field does not change with time, the system is in an electrostatic state, and the resulting field is an electrostatic field. The field produced by a fixed point charge$Q$is

$\mathbf E=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}\hat{\mathbf r}$

Here, "static" refers to the physical state; "conservative" refers to the line-integral property of the field. These are not the same definition. An important property of the electrostatic field is that, in an ordinary simply connected region, the integral along any closed path satisfies

$\oint\mathbf E\cdot d\mathbf l=0,\qquad \nabla\times\mathbf E=0$

A zero closed-loop integral means that the integral from$A$to$B$does not depend on which path is taken. Therefore, we can define the electric potential difference

$V_B-V_A=-\int_A^B\mathbf E\cdot d\mathbf l$

After choosing a reference zero point, every position has a corresponding electric potential, and

$\mathbf E=-\nabla V$

The order of this reasoning is important: it is not that a "potential slope" exists first and then the charge experiences a force; rather, the path independence of the electrostatic field allows us to describe it with a scalar potential. The work done by the electric field force on a positive charge satisfies$W_{A\to B}/q=V_A-V_B$.

What kind of electric field does a changing magnetic field produce?

Faraday discovered that even without a battery, as long as the magnetic flux through a fixed closed loop changes with time, a current may appear in the loop. For a fixed loop, the experimental law is written as

$\oint\mathbf E\cdot d\mathbf l=-\frac{d\Phi_B}{dt}$

The corresponding local equation is the Maxwell–Faraday equation:

$\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}$

If the rate of change of magnetic flux enclosed by a loop is nonzero, the circulation of the electric field along that loop is nonzero; such a total electric field cannot be described by a single-valued scalar potential alone. The part produced by a time-varying magnetic field is called the induced electric field. It is not a "fictitious non-electric field": charges really do experience the effect of$q\mathbf E_{\mathrm{ind}}$. In real situations, the electric field produced by the charge distribution and the induced electric field can exist simultaneously, and a general electric field cannot simply be divided into two unrelated categories.

For a general time-varying electromagnetic field, one can introduce a scalar potential$\phi$and a vector potential$\mathbf A$:

$\mathbf E=-\nabla\phi-\frac{\partial\mathbf A}{\partial t}$

The scalar potential can still be defined, but by itself it is not enough to determine the total electric field. Note also that the specific forms of$\phi$and$\mathbf A$change with the choice of gauge; what is physically measurable is the complete electric and magnetic fields. Therefore, the two terms in the formula should not be described under all circumstances as uniquely and independently measurable "conservative" and "non-conservative" fields.

Why can't the electrostatic force keep a charge moving around a loop indefinitely?

When a charge is in an electrostatic field, the electrostatic force it experiences is

$\mathbf F_{\mathrm{es}}=q\mathbf E_{\mathrm{es}}$

Because the electrostatic field is conservative, the total work done by the electrostatic force along a closed loop is zero. With only this force, a charge does not gain net energy after completing one lap.

Take a DC circuit consisting of a battery and an external circuit as an example: the electric field in the external circuit drives positive charges from a higher electric potential to a lower electric potential; after the charges enter the battery, the chemical processes inside the battery maintain charge separation and send the positive charges from the low-potential end back to the high-potential end. Without continuous energy supply inside the battery, the potential difference in the circuit would gradually disappear, and a steady current could not be maintained.

In the context of middle school circuits, the mechanism inside a battery that overcomes electrostatic effects and replenishes energy to charges is often called a "non-electrostatic force." Accordingly, electromotive force represents the work done by the internal action of the source on unit positive charge:

$\mathcal E=\frac{W_{\mathrm{源}}}{q}$

Its unit is$\mathrm{J/C}=\mathrm V$. Although the term electromotive force contains the word "force," it is not actually a kind of force, nor is it equivalent to the electric potential difference between two points; it answers: how much energy can the source provide for each coulomb of positive charge it transports? In a real circuit, the terminal voltage is also affected by factors such as internal resistance.

"Non-electrostatic force" is not another unified kind of field

This term must be understood in a specific context. When discussing batteries, it usually refers to the chemical action inside the source; when discussing electromagnetic induction, what drives charges in a fixed loop can be the induced electric field; when discussing a moving conductor, charges also experience the Lorentz force

$\mathbf F=q(\mathbf E+\mathbf v\times\mathbf B)$

where$q\mathbf v\times\mathbf B$is the magnetic force, not the electric field force. The magnetic force in a moving conductor can separate charges and produce a motional electromotive force, but the magnetic force is always perpendicular to the instantaneous velocity of the charge and itself does no work on free charges; the energy obtained by the circuit comes from the mechanical external force that keeps the conductor moving.

Therefore, the induced electric field, the chemical action in a battery, and the magnetic force in a moving conductor do not have the same physical mechanism. They may all participate in establishing or driving electromotive force, but they should not therefore be regarded as the same kind of "non-electrostatic field." Different textbooks may also use "non-electrostatic force" over different ranges; when discussing batteries, it is safest to understand it as "the mechanism inside the source that overcomes electrostatic effects."

Putting the concepts back in their proper places

• Electrostatic fieldDescribes the electric field state when the charge distribution does not change with time; in ordinary regions, it is a conservative field and can be fully described by electric potential.

• Induced electric fieldProduced by a changing magnetic field and still a real electric field; if its closed-loop circulation is nonzero, the total field cannot be described by a single-valued scalar potential alone.

• Electrostatic forceIs the effect of an electrostatic field on a charge, and its net work along a closed path is zero.

• Electromotive forceIs the energy gained per unit charge, not a kind of force; the energy-supplying mechanisms corresponding to batteries, changing magnetic fields, and moving conductors are all different.

Remember these three questions, and it becomes hard to mix up the terminology: What field is present in space? What force does a charge experience? Who supplies the energy per unit charge?

References: [OpenStax University Physics: Electrostatic Force and Electric Potential Energy](https://openstax.org/books/university-physics-volume-2/pages/7-1-electric-potential-energy); [OpenStax: Faraday's Law](https://openstax.org/books/university-physics-volume-2/pages/13-1-faradays-law); [MIT OpenCourseWare: Faraday's Law and Induced Electric Field](https://ocw.mit.edu/courses/8-02t-electricity-and-magnetism-spring-2005/724a162b8c03487f5faae202b395fadd_cha10faraday_law.pdf).