vector field

(noun)

a construction in which each point in a Euclidean space is associated with a vector; a function whose range is a vector space

Related Terms

Examples of vector field in the following topics:

  • Superposition of Fields

    • As vector fields, electric fields obey the superposition principle.
    • Possible stimuli include but are not limited to: numbers, functions, vectors, vector fields, and time-varying signals.
    • Electric fields are continuous fields of vectors, so at a given point, one can find the forces that several fields will apply to a test charge and add them to find the resultant.
    • To do this, first find the component vectors of force applied by each field in each of the orthogonal axes.
    • Then once the component vectors are found, add the components in each axis that are applied by the combined electric fields.

  • Electric Field Lines: Multiple Charges

    • Electric fields created by multiple charges interact as do any other type of vector field; their forces can be summed.
    • Each will have its own electric field, and the two fields will interact.
    • More field lines per unit area perpendicular to the lines means a stronger field.
    • As vector fields, electric fields exhibit properties typical of vectors and thus can be added to one another at any point of interest.
    • Thus, given charges q1, q2 ,... qn, one can find their resultant force on a test charge at a certain point using vector addition: adding the component vectors in each direction and using the inverse tangent function to solve for the angle of the resultant relative to a given axis.

  • Helical Motion

    • Helical motion results when the velocity vector is not perpendicular to the magnetic field vector.
    • In the section on circular motion we described the motion of a charged particle with the magnetic field vector aligned perpendicular to the velocity of the particle.
    • In this case, the magnetic force is also perpendicular to the velocity (and the magnetic field vector, of course) at any given moment resulting in circular motion.
    • shows how electrons not moving perpendicular to magnetic field lines follow the field lines.
    • When a charged particle moves along a magnetic field line into a region where the field becomes stronger, the particle experiences a force that reduces the component of velocity parallel to the field.

  • Tensors

    • Because transforms as a contravariant vector and doesn't transform, must transform as a covariant vector.
    • We could also imagine taking the derivative of the vector field to create a tensor, for example,
    • We have argued that we can only measure the fields themselves, so we would like to figure out how the fields transform.
    • Under rotations the fields act like vectors.
    • Can we generalize the electric and magnetic field to be four-vectors?

  • Constant Velocity Produces a Straight-Line

    • Because velocity is a vector, the direction remains unchanged along with the speed, so the particle continues in a single direction, such as with a straight line.
    • The force a charged particle "feels" due to a magnetic field is dependent on the angle between the velocity vector and the magnetic field vector B .
    • In this case a charged particle can continue with straight-line motion even in a strong magnetic field.
    • In the case above the magnetic force is zero because the velocity is parallel to the magnetic field lines.
    • Identify conditions required for the particle to move in a straight line in the magnetic field

  • Electric vs. Magnetic Forces

    • Where F is the force vector, q is the charge, and E is the electric field vector.
    • A consequence of this is that the electric field may do work and a charge in a pure electric field will follow the tangent of an electric field line.
    • where B is the magnetic field vector, v is the velocity of the particle and θ is the angle between the magnetic field and the particle velocity.
    • The angle dependence of the magnetic field also causes charged particles to move perpendicular to the magnetic field lines in a circular or helical fashion, while a particle in an electric field will move in a straight line along an electric field line.
    • The electric field is directed tangent to the field lines.

  • Superposition of Forces

    • Therefore, the principle suggests that total force is a vector sum of individual forces.
    • The resulting force vector happens to be parallel to the electric field vector at that point, with that point charge removed.
    • where qi and ri are the magnitude and position vector of the i-th charge, respectively, and $\boldsymbol{\widehat{R_i}}$ is a unit vector in the direction of $\boldsymbol{R}_{i} = \boldsymbol{r} - \boldsymbol{r}_i$ (a vector pointing from charges qi to q. )
    • For example, when a charge is moving in the presence of a magnetic field as well as an electric field, the charge will feel both electrostatic and magnetic forces.
    • The E field and B field vary in space and time.

  • Linear Vector Spaces

    • We can add any two, say, force vectors and get another force vector.
    • We can also scale any such vector by a numerical quantity and still have a legitimate vector.
    • Definition of Linear Vector Space: A linear vector space over a field $F$ of scalars is a set of elements $V$ together with a function called addition from $V \times V$ into $V$ (the Cartesian product $A \times B$ of two sets $A$ and $B$ is the set of all ordered pairs $(a,b)$ where $a \in A$ and $b \in B$ ) and a function called scalar multiplication from $F \times V$ into $V$ satisfying the following conditions for all $x,y,z \in V$ and all $\alpha, \beta \in F$ :
    • The simplest example of a vector space is $\mathbf{R}^n$ , whose vectors are n-tuples of real numbers.
    • In the case of $n=1$ the vector space $V$ and the field $F$$F$ are the same.

  • Solving Problems with Vectors and Coulomb's Law

    • Coulomb's Law using vectors can be written as:
    • This means that we need to subtract the corresponding components of vector $\mathbf{r}_Q$ from vector $\mathbf{r}_q$.
    • Electric Force on a Field Charge Due to Fixed Source Charges
    • The total force on the field charge q is due to applications of the force described in the vector notation of Coulomb's Law from each of the source charges.
    • The displacements of the field charge from each source charge are shown as light blue arrows.

  • Ampere's Law: Magnetic Field Due to a Long Straight Wire

    • Current running through a wire will produce a magnetic field that can be calculated using the Biot-Savart Law.
    • Current running through a wire will produce both an electric field and a magnetic field.
    • In this equation, the r vector can be written as r̂ (the unit vector in direction of r), if the r3 term in the denominator is reduced to r2 (this is simply reducing like terms in a fraction).
    • As illustrated in the direction of the magnetic field can be determined using the right hand rule—pointing one's thumb in the direction of current, the curl of one's fingers indicates the direction of the magnetic field around the straight wire.
    • The direction of the magnetic field can be determined by the right hand rule.