Electric Fields in Uniform and Nonuniform Regions

Charged Particles in a Uniform Electric Field

When a charged particle is placed between the plates of a parallel plate capacitor, it experiences a uniform electric field. The behavior of the particle depends on the direction and magnitude of the field, as well as the sign of the charge.

• Electric Field Direction: The electric field inside a parallel plate capacitor points from the positive plate to the negative plate.

• Force on a Charge: The force on a positive charge is in the direction of the electric field; for a negative charge, it is opposite to the field.

• Acceleration: The acceleration of the particle is determined by the net force acting on it, according to Newton's second law: .

• Motion: If the force and velocity are in opposite directions, the particle slows down; if they are in the same direction, it speeds up.

Example: A negatively charged particle in a uniform field directed to the right will experience a force to the left, causing it to slow down if it was initially moving to the right.

Electric Dipoles in Electric Fields

Definition and Properties of an Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a fixed distance. The dipole moment is a vector pointing from the negative to the positive charge, with magnitude .

• Dipole Moment: , where is the charge and is the displacement vector from negative to positive charge.

Torque on a Dipole in a Uniform Electric Field

When placed in a uniform electric field , a dipole experiences a torque that tends to align the dipole moment with the field.

• Torque (vector form):

• Torque (magnitude):

• Angle : The angle between and .

Example: If is perpendicular to , the torque is maximized.

Dipole Behavior in Uniform and Nonuniform Fields

• Uniform Field: The net force on the dipole is zero, but there is a torque that rotates the dipole to align with .

• Nonuniform Field: The dipole experiences both a net force and a torque. The net force moves the dipole toward regions of stronger field (typically in the direction of the force on the positive charge).

Example: In a nonuniform field where the field is stronger on one side, the dipole will move toward that side.

Comparison of Torques on Dipoles

Magnitude of Torque in Different Field Directions

The magnitude of the torque on a dipole in a uniform electric field depends on the angle between and , but not on the direction of itself.

• Formula:

• Symmetry: , so reversing the field direction does not change the torque magnitude.

Example: If two identical dipoles are placed in fields of equal magnitude but opposite direction, and at the same angle, the torque magnitudes are equal.

Applications: Microwave Cooking and Dipoles

Microwave Interaction with Water Molecules

Microwave ovens produce rapidly oscillating electric fields. Water molecules, which are permanent dipoles, experience torques in these fields, causing them to rotate and transfer energy to surrounding molecules, resulting in heating.

• Water as a Dipole: Water molecules have a permanent dipole moment due to their molecular structure.

• Heating Mechanism: The oscillating field causes water molecules to rotate, increasing thermal energy.

Example: Placing metal in a microwave can cause sparks due to the free electrons in the metal responding to the oscillating field.

Symmetry and Electric Fields

Symmetry in Charge Distributions

The symmetry of a charge distribution determines the symmetry of the resulting electric field. Recognizing these symmetries helps in applying Gauss's law and predicting field patterns.

• Cylindrical Symmetry: Invariant under translation along the axis, rotation about the axis, and reflection in planes containing the axis.

• Planar Symmetry: Invariant under translation parallel to the plane, rotation about an axis perpendicular to the plane, and reflection in the plane.

• Spherical Symmetry: Invariant under rotation about any axis through the center and reflection in any plane containing the center.

Electric Field of a Cylinder of Charge

An infinitely long, uniformly charged cylinder aligned with the x-axis exhibits cylindrical symmetry. The electric field must share this symmetry.

• Field Direction: The field points radially outward from the axis of the cylinder.

• Dependence: The field depends only on the distance from the axis, not on the position along the axis or the angle around it.

Electric Field of a Plane of Charge

A uniformly charged infinite plane (e.g., in the xz-plane at y = 0) exhibits planar symmetry. The electric field must be perpendicular to the plane and uniform in magnitude.

• Field Direction: The field points away from the plane if the charge is positive, toward the plane if negative.

• Uniformity: The field is the same at all points equidistant from the plane.

Summary Table: Dipole Behavior in Electric Fields

Key Equations

• Force on a charge:

• Torque on a dipole:

• Magnitude of torque:

Additional info: Some context and explanations have been expanded for clarity and completeness, including the summary table and explicit definitions of symmetry types.