Scalar Fields Scalar Fields: If we assign a number to each location in space, we have a scalar field. An other way of putting it is to assign a function: G Sxyz( ,,) = Sx( ) to each point in space. Properties: We usually deal with scalar fields that are “well behaved”, i.e. no singularities (infinites), and no abrupt discontinuities (sudden jumps.) Examples: Temperature, pressure, gravitational potential, electric potential,…

Scalar Field: Ocean charts

1 Vector Fields Vector Fields: If we assign a vector (3 numbers) to each location in space, we have a vector field. An other way of putting it is to assign a function: G G Vx( ) to each point in space. Properties: We usually deal with vector fields that are “well behaved”, i.e. no singularities (infinites), and no abrupt discontinuities (sudden jumps) no sudden breaks in direction. Examples: Velocities in a fluid, E-field, B-field.

Vector Fields examples: Sea-Ice motion from radar.

Meteorology example (Swiss)

2 Electric Field The electric field is a vector field that is defined by the force divided by the charge that a small test charge would experience at any point in space: G G G G Fx( ) Ex= N  ()  C q0 Electric fields are “real”, since they can store energy (as we will see later.)

Special Properties of E-fields: All the E-field vectors are along the tangents of lines, which we call the electric field lines. The electric field lines all start on a positive charge and end on a negative charge. The electric field lines can never cross. For conductors, the electric field lines always start or end perpendicular to the conducting surface.

Superposition for Electric Fields

For multiple charges, the electric fields will add as a vector sum: GG GG G EE=+++=012 EE... ∑ Ei

3 E-field due to a point charge

1 qq12 F = 2 4πε 0 r 1 q E = 2 4πε 0 r

Electric Field for a dipole

11qq EE=−=()+− E () 22 − 44πε00rr+ πε − 12qd 1 p E ==33 42πε00zz πε G G pqd= []Cm It is customary to have the direction of the p vector point from the – to the + charge.

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