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#### Notes by Category University Engineering

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# Electric Fields

Electric fields are regions where non-contact electrical forces can be felt by charged objects. They are generated by electrical charges.

Field lines show the path a positive test charge would take from positive to negative. The closer they are, the stronger the field. Lines never stop in empty space and must never cross. Field lines go from positive to negative, so the point charge below is negative: Uniformly charged spheres can be modelled as a point charge at its centre, with field lines leaving/entering at right angles all around, to infinity. Closer to the point, you can see that the field lines are closer together. Therefore, the field strength must be greater. In all radial fields, the field strength is proportional to the distance from the point charge via an inverse square law.

Electric field strength is force per unit charge:

E = F/Q Electric field strength = Force / Charge

The units of electric field strength and newtons per coulomb, N/C

## Coulomb's Law

The attractive/repulsive force between two point charges can be calculated using Coulomb's law:

F = Qq / 4πεr² Force = Product of two charges / 4π x ε x separation²

ε is the permittivity of free space, and can be taken as 8.85 E-12.

Since electric field strength is calculated as E = F/Q, coulomb's law can be used to calculate field strength of a point charge by dividing by one charge:

F = Q / 4πεr² Force = Charge / 4π x ε x separation²

### Electric vs Gravitational Fields

Newton's law of gravitation is very similar to coulomb's law, and the fields have many other similarities, too. However, there are some differences.

Similarities

• Point masses/charges both produce radial fields

• Force and field strength are inversely proportional to distance²

• Force is proportional to product of masses/charges

• Newton's and Coulomb's laws are in same format, with different coefficients and mass and charge respectively

Differences

• Electric fields can be attractive or repulsive; gravitational is always attractive

• Mass produces the field for gravitational fields, charges for electrical

## Uniform Electric Fields

In a uniform electric field, the field strength is the same everywhere: E = V/d Field strength = p.d. / distance

The field lines are parallel to one another and evenly spaced.

### Capacitors & Electric Fields

Parallel plate capacitors work by using a uniform electric field across the insulator between the two plates. Remember that the charge on each plate is equal but opposite.

There are two basic types of parallel plate capacitors, those with a vacuum between the two plates as the insulator, and those with an insulating material (dielectric).

• For capacitors with a vacuum, the permittivity of free space, ε₀, and the area, A, of the plates (just one plate - the areas are the same) is required to calculate capacitance:

C = ε₀A/d capacitance = permittivity of free space x area / plate separation
• For capacitors with a dielectric, the permittivity, ε, of that material must be calculated: ε = ε₀ x ε(r). ε(r) is the relative permittivity. Therefore, capacitance is:

C = εA/d capacitance = permittivity x area / plate separation

### Motion of Charged Particles in Uniform Electric Fields

Charged particles in a uniform electric field accelerate towards the oppositely charged plate at a uniform rate. Charged particles moving perpendicular to the field move in a curved shape, just like projectile motion. This is because a particle of charge Q experiences a constant force (F = EQ) acting parallel to the field lines. The work done increases, and so does its kinetic energy. This means it accelerates at a uniform rate. If the velocity of the particle has components perpendicular to the field lines, this will remain unchanged, resulting in a curved path.

## Electrical Potential and Energy

Electric potential is the work done in bringing the unit charge from infinity to a point. Electric potential at infinity is 0.

In a radial field, electric potential is given as

V = Q/4πε₀r Electrical potential = charge / 4π x ε₀ x distance

The units of electric potential are volts.

• When Q is positive, so is V: the force is repulsive.

• When Q is negative, so is V: the force is attractive. The absolute magnitude of V is greatest at the surface of the charge and decreases as the distance increases. Graphs of force against distance look the same, but the area under the graph is equal to the work done in moving the unit charge.

Therefore, electrical potential energy (the work done) is given as:

E = Vq Electrical potential energy = electrical potential x charge
E = Qq/4πε₀r Electrical potential energy = product of charges / 4π x ε₀ x separation

### Electrical Potential and Capacitance

Since capacitance is given as C = Q/V, we can substitute in the equation for electrical potential, V, to get the capacitance of an isolated sphere:

C = 4πε₀R Capacitance = 4π x ε₀ x radius of the sphere

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