Electric Field Strength

Divide the applied voltage by the plate separation: E = U/d. With U = 200 V and d = 4 mm = 0.004 m you get E = 50,000 V/m. The field between the plates is uniform: it has the same magnitude and direction everywhere, from the positive to the negative plate. The force on a charge then follows from F = q·E; an electron here experiences F = 1.6×10⁻¹⁹ × 5×10⁴ = 8×10⁻¹⁵ N. The most important error source is the plate separation in millimetres: d must be entered in metres, otherwise E is off by a factor of 1000.

E = F/q is the definition of field strength and holds without exception in every electric field: imagine a small test charge placed at the point and divide the force by the charge. E = U/d, by contrast, is a special formula valid only in a uniform field, in practice between the plates of a capacitor away from the edges. There the potential decreases linearly with position, so the quotient of voltage and distance is constant. In the field of a point charge E = U/d would be wrong; there E = k_e·Q/r² applies and the field falls off quadratically. Remember: the definition always, the plate formula only for uniform fields.

Yes, both are exactly equivalent and reflect the two faces of field strength. From the force definition E = F/q the unit N/C follows. From the capacitor formula E = U/d follows V/m. The conversion is pure unit algebra: 1 V = 1 J/C = 1 N·m/C, hence 1 V/m = 1 N·m/(C·m) = 1 N/C. Both views are useful: N/C emphasises that the field exerts forces on charges; V/m emphasises that the field represents a potential gradient, i.e. voltage per distance. In problems you may switch freely between the two, and unit checks work equally well in either representation.

The electron experiences the constant force F = e·E, opposite to the field direction because its charge is negative. Constant force means constant acceleration a = e·E/m, so the electron moves like a projectile in a gravitational field. Flying parallel to the field it is uniformly accelerated or decelerated; the energy balance gives the practical formula ½mv² = e·U. Entering perpendicular to the field, for instance between the deflection plates of an oscilloscope, uniform longitudinal motion and accelerated transverse motion superpose into a parabolic path, mathematically identical to a horizontal launch. Gravity is negligible in comparison, since the electric force exceeds the electron weight by many orders of magnitude.

From about 3 kV/mm (3×10⁶ V/m) air becomes conductive: the breakdown field strength is reached. Free electrons, always present through natural ionisation, are accelerated so strongly in the field that they ionise air molecules on impact and release further electrons, creating an avalanche. This becomes visible as a spark or arc, on the largest scale as lightning. Practical consequences: between capacitor plates 1 mm apart at most about 3 kV is possible, and high-voltage lines need large clearances and smooth, large radii of curvature, because the field is locally enhanced at sharp points (point effect, used in lightning rods). This limit also explains the crackling of high-voltage equipment (corona discharge).