Physics (9702) | Electric fields

A Level Physics 9702 Study Notes: Chapter 18 – Electric Fields

Welcome to the world of Electric Fields! This chapter is crucial because it takes the AS Level concepts of charge and current and dives deeper into the forces and energy associated with static charges. Think of electric fields as the "how" behind the electrical forces you’ve already studied. Understanding these concepts is essential for moving on to Capacitance (Chapter 19) and future advanced topics.

Don't worry if fields seem abstract! We will use analogies to gravity, which behaves very similarly, to make the concepts clear.

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18.1 Electric Fields and Field Lines

What is an Electric Field?

An electric field is an example of a field of force. Just like a gravitational field surrounds a mass, an electric field surrounds an electric charge.

The field is the region where an electric charge experiences a force.

Definition: The Electric Field Strength (\(E\)) at a point is defined as the force per unit positive charge experienced by a small test charge placed at that point.

• Electric Field Strength (\(E\)) is a vector quantity (it has both magnitude and direction).
• The unit of \(E\) is Newtons per Coulomb (\(\text{N C}^{-1}\)) or, as we will see later, Volts per metre (\(\text{V m}^{-1}\)).

The Force Equation (\(F = qE\))

If you know the electric field strength \(E\) at a point, you can calculate the force \(F\) exerted on any charge \(q\) placed there:

$$F = qE$$

• If \(q\) is positive, the force \(F\) is in the same direction as the field \(E\).
• If \(q\) is negative, the force \(F\) is in the opposite direction to the field \(E\).

Analogy Check: This is exactly like gravitational fields, where \(g\) is the field strength (force per unit mass) and the weight \(W\) is the force on a mass \(m\). \(W = mg\) is the mechanical equivalent of \(F = qE\).

Representing Electric Fields (Field Lines)

Electric fields are represented by lines called field lines or lines of force. These lines show the direction a small positive test charge would move if placed in the field.

Rules for Drawing Field Lines:

1. They start on positive charges and end on negative charges (or at infinity).
2. The direction of the line gives the direction of the force on a positive charge.
3. The density (closeness) of the lines indicates the strength of the field: closer lines mean a stronger field.
4. Field lines never cross each other.

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18.2 Uniform Electric Fields

What is a Uniform Field?

A uniform electric field is one where the field strength \(E\) is the same (in both magnitude and direction) at all points in that region. This is typically created by placing two large, parallel, conducting plates close to each other and connecting them to a power source.

Between the plates, the field lines are parallel, equally spaced, and perpendicular to the plates.

Field Strength and Potential Difference (\(E = \Delta V / \Delta d\))

In a uniform field, the electric field strength \(E\) is related to the potential difference (\(\Delta V\)) between the plates and the separation distance (\(\Delta d\)).

$$E = \frac{\Delta V}{\Delta d}$$

where:

• \(\Delta V\) is the potential difference (Voltage) in Volts (V).
• \(\Delta d\) is the separation distance between the plates in metres (m).
• \(E\) is the electric field strength in Volts per metre (\(\text{V m}^{-1}\)).

Did you know? This confirms that the units \(\text{N C}^{-1}\) and \(\text{V m}^{-1}\) are equivalent!

Motion of Charged Particles in a Uniform Field

When a charged particle (like an electron or proton) enters a uniform electric field perpendicular to the field lines, it experiences a constant force (\(F=qE\)).

Since the force is constant, the particle undergoes a constant acceleration, \(a = F/m = qE/m\). The motion is exactly analogous to a projectile moving in a uniform gravitational field (like near the Earth's surface).

Key aspects of the motion:

1. The particle maintains a constant velocity component parallel to the plates (since no force acts in that direction).
2. The particle undergoes constant acceleration perpendicular to the plates (in the direction of the electric force).
3. The resulting path is parabolic.

Practical Use: This principle is used in devices like old cathode-ray oscilloscopes (CROs) to deflect electron beams and draw images on the screen.

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18.3 Electric Force between Point Charges (Coulomb’s Law)

The calculation of force between static point charges is governed by Coulomb's Law. This law is fundamental and has the same inverse square dependence as Newton's Law of Gravitation.

Coulomb's Law

The electric force \(F\) between two point charges, \(Q_1\) and \(Q_2\), separated by a distance \(r\) in free space (vacuum), is proportional to the product of the charges and inversely proportional to the square of the separation distance.

$$F = \frac{Q_1 Q_2}{4 \pi \epsilon_0 r^2}$$

where:

• \(Q_1\) and \(Q_2\) are the magnitudes of the charges (in C).
• \(r\) is the separation distance (in m).
• \(\epsilon_0\) is the permittivity of free space (a constant provided in your data booklet).
• The term \(1/(4 \pi \epsilon_0)\) is sometimes written as the Coulomb constant, \(k\).

Nature of the Force:

• If \(Q_1\) and \(Q_2\) have the same sign (like charges), \(F\) is positive, meaning the force is repulsive.
• If \(Q_1\) and \(Q_2\) have opposite signs (unlike charges), \(F\) is negative, meaning the force is attractive.

Important Note on Conductors (Syllabus point): For a point outside a spherical conductor, the charge on the sphere may be considered to be a point charge at its centre. This simplifies force and field calculations immensely, allowing us to use Coulomb's Law even if one object isn't strictly a point.

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18.4 Electric Field of a Point Charge

If we want to find the field strength \(E\) generated by a single charge \(Q\) at a distance \(r\), we combine the definition of field strength (\(E = F/q\)) with Coulomb's Law.

Imagine the generating charge is \(Q\) and the small test charge is \(q\). The force is \(F = \frac{Q q}{4 \pi \epsilon_0 r^2}\). Dividing by \(q\):

$$E = \frac{Q}{4 \pi \epsilon_0 r^2}$$

This equation tells us:

• The electric field strength \(E\) decreases rapidly as \(r\) increases (due to the \(1/r^2\) dependence).
• The direction of \(E\) is radially outward from a positive charge and radially inward towards a negative charge.

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18.5 Electric Potential (\(V\))

Potential is perhaps the trickiest concept, but it is necessary because electric fields do work on charges. Potential is a way of mapping the energy landscape of the field.

Defining Electric Potential (The Work Done)

Electric Potential (\(V\)) at a point is defined as the work done per unit positive charge in bringing a small test charge from infinity to that point.

• Potential is a scalar quantity (only magnitude, no direction).
• The unit is Joules per Coulomb (\(\text{J C}^{-1}\)), which is equivalent to the Volt (V).
• Infinity is defined as having zero electric potential (\(V=0\)).

Encouraging Note: Because potential is a scalar, calculations involving multiple charges are much easier! You just add or subtract the potentials algebraically (paying attention to the sign of the charge Q).

Electric Potential Due to a Point Charge

For a charge \(Q\) in free space, the electric potential \(V\) at a distance \(r\) is given by:

$$V = \frac{Q}{4 \pi \epsilon_0 r}$$

Notice the critical difference compared to the field strength equation: Potential is proportional to \(1/r\), not \(1/r^2\).

• Positive charges create positive potential (like hills on an energy map).
• Negative charges create negative potential (like valleys on an energy map).

Electric Potential Energy (\(E_p\))

If the potential at a point is \(V\), and you place a charge \(q\) at that point, the electric potential energy (\(E_p\)) stored in the system is:

$$E_p = qV$$

Substituting the potential formula (\(V\)), the potential energy of two point charges \(Q\) and \(q\) separated by distance \(r\) is:

$$E_p = \frac{Q q}{4 \pi \epsilon_0 r}$$

Interpreting Potential Energy:

• If \(E_p\) is positive (both charges are the same sign), the charges repel. Work had to be done against the repulsive force to bring them together.
• If \(E_p\) is negative (opposite signs), the charges attract. Work would have to be done to separate them. A negative potential energy indicates a stable, bound system.

Electric Field and Potential Gradient (\(E = -dV/dr\))

The electric field \(E\) is related to how rapidly the electric potential \(V\) changes over distance \(r\). This relationship is called the potential gradient.

$$E = - \frac{dV}{dr}$$

What does this mean?

• \(dV/dr\) is the rate of change of potential with distance (the gradient).
• The magnitude of the Electric Field Strength \(E\) is equal to the magnitude of the potential gradient.
• The negative sign (\(-\)) is crucial: it shows that the electric field vector \(E\) points in the direction of the steepest decrease in electric potential. Charges naturally accelerate from high potential to low potential (like water flowing downhill).