Inphysical theories, atest particle, ortest charge, is an idealized model of an object whose physical properties (usuallymass,charge, orsize) are assumed to be negligible except for the property being studied, which is considered to be insufficient to alter the behaviour of the rest of the system. The concept of a test particle often simplifies problems, and can provide a good approximation for physical phenomena. In addition to its uses in the simplification of the dynamics of a system in particular limits, it is also used as a diagnostic incomputer simulations of physical processes.

In simulations withelectric fields the most important characteristics of a test particle is itselectric charge and itsmass. In this situation it is often referred to as atest charge.

The electric field created by a point chargeq is

${\displaystyle \mathbf{\text{E}}=\frac{q\hat{\mathbf{r}}}{4\pi {\epsilon }_0{r}^2}}$,

whereε₀ is thevacuum electric permittivity.

Multiplying this field by a test charge${\displaystyle q_{\text{test}}}$ gives an electric force (Coulomb's law) exerted by the field on a test charge. Note that both theforce and theelectric field are vector quantities, so a positive test charge will experience a force in the direction of the electric field.

The easiest case for the application of a test particle arises inNewton's law of universal gravitation. The general expression for the gravitational force between any two point masses${\displaystyle m_1}$ and${\displaystyle m_2}$ is:

${\displaystyle F=-G\frac{m_1{m}_2}{|{\mathbf{r}}_1-{\mathbf{r}}_2{|}^2}}$,

where${\displaystyle {\mathbf{r}}_1}$ and${\displaystyle {\mathbf{r}}_2}$ represent the position of each particle in space. In the general solution for this equation, both masses rotate around theircenter of massR, in this specific case:^[1]

${\displaystyle \mathbf{R}=\frac{m_1{\mathbf{r}}_1+m_2{\mathbf{r}}_2}{m_1+m_2}}$.

In the case where one of the masses is much larger than the other (${\displaystyle m_1\gg m_2}$), one can assume that the smaller mass moves as a test particle in agravitational field generated by the larger mass, which does not accelerate. We can define the gravitational field as

${\displaystyle \mathbf{g}(r)=-\frac{G{m}_1}{r^2}\hat{\mathbf{r}}}$,

with${\displaystyle r}$ as the distance between the massive object and the test particle, and${\displaystyle \hat{r}}$ is the unit vector in the direction going from the massive object to the test mass.Newton's second law of motion of the smaller mass reduces to

${\displaystyle \mathbf{a}(r)=\frac{F}{m_2}\hat{\mathbf{r}}=\mathbf{g}(r)}$,

and thus only contains one variable, for which the solution can be calculated more easily. This approach gives very good approximations for many practical problems, e.g. the orbits ofsatellites, whose mass is relatively small compared to that of theEarth.

In metric theories of gravitation, particularlygeneral relativity, a test particle is an idealized model of a small object whose mass is so small that it does not appreciably disturb the ambientgravitational field.

According to theEinstein field equations, the gravitational field is locally coupled not only to the distribution of non-gravitationalmass–energy, but also to the distribution ofmomentum andstress (e.g. pressure, viscous stresses in aperfect fluid).

In the case of test particles in avacuum solution orelectrovacuum solution, this turns out to imply that in addition to the tidal acceleration experienced by small clouds of test particles (with spin or not), test particles withspin may experience additionalaccelerations due tospin–spin forces.^[2]

• Point particle (point mass,point charge)

• Mathematical methods in general relativity