Electric field strength
1. Electric Field line:
– Electric field lines emerge from positive charges (sources) and enter negative charges (sinks).
Figure 1 Electric Field line (a) different charges
(b) Same charges
- – The direction of the lines indicates the direction of the electric field.
– The density of the lines represents the magnitude of the electric field.
– Electric field lines never intersect or cross each other.
– Electric field lines are a model to help us visualize a field, but a direct way of showing an electric field is shown in Figure 2.
– In this photograph, a potential difference has been applied to two metal plates, which have been placed into an insulating liquid.
– Then short pieces of a fine thread have been sprinkled on top of the liquid.
– When the electric field is applied, the pieces of thread line up along the field lines, in the same way that iron filings follow magnetic field lines. 
Figure 2 Electric field- – Electric field lines are continuous and unbroken.
– They form a vector field, with both magnitude and direction.
– They can be used to visualize and analyze electric fields around various charge configurations. - Applications:
– Understanding electric field strength and direction.
– Analyzing electric field patterns around charges and charge distributions.
– Visualizing electric fields in various environments (e.g., around conductors, insulators, and electrodes).
2. Electric field strength:
- Two-point charges exerting a force on each other.
- A charge produces an electric field around it, which exerts a force on another charged object.
- This idea is similar to a magnetic field close to a magnet, or a gravitational field around a planet.
- Electric field strength (E) is the magnitude of the electric field at a given point in space.
- It is a vector quantity, typically measured in units of Newtons per Coulomb (N/C) or Volts per meter (V/m).
- [math] E = \frac{F}{Q} [/math]
- – E is the electric field strength (N/C or V/m)
- – F is the force exerted on a test charge (N)
– Q is the magnitude of the test charge (C) - The direction of the electric field is defined as the direction of the force on a positive charge.
- Electric field is a vector quantity because it has both magnitude and direction.
Figure 3 Electric field strength- Physical Interpretation:
– The electric field strength (E) represents the intensity of the electric field at a given point.
– It is a vector quantity, with both magnitude and direction.
– The direction of E is the same as the direction of the force on a positive test charge.
– E is a radial vector, decreasing in magnitude with increasing distance from the source charge. - Application:
- Electric field strength is crucial in understanding various phenomena, such as:
– Electric forces and torques
– Electric potential and potential difference
– Capacitance and capacitors
– Electric currents and resistance
– Electromagnetic induction and waves -
Example:
A small charge of +2µC is placed in the electric field in figure 4. What force does it experience?
Figure 4 Electric field for 400 N/C- Given data:
- Electric field [math] = E = 400 N/C [/math]
- Charge [math] = Q = +2µC = 2 * 10^{-6} C [/math]
Applied force = F=?
Formula: - [math] E = \frac{F}{Q} \\ F = EQ [/math]
- Solution:
- [math] F = EQ \\ F = (400)(2 *10^{-6}) \\ F = 8 *10^{-4} \, \text{N (downwards)} [/math]
3. Uniform electric field:
- Figure 2 shows a charge +Q placed in an electric field between two parallel plates.
- The plates have a potential difference of V between them, and their separation is d (in m).
- First, the work done = F * d, but the force to move the charge must be equal in magnitude to the force on the charge, due to the electric field = E * Q. So
- work done = EQd
- Secondly, the work done is also equal to the energy gained by the charge in moving through a potential difference V.
- This is VQ – you should remember that a volt is defined as a joule per coulomb. Therefore
-
EQd = VQ
- and
- [math] E = \frac{V}{d} [/math]
- This equation allows us to calculate the magnitude of a uniform electric field between two parallel plates. Note that the electric field strength can also be measured in [math]V.m^{-1}[/math].
- Example:
- Two parallel plates are separated by a distance of 2 cm (0.02 m). The potential difference between the plates is 100 V. Find the electric field strength (E) between the plates.
- Given data:
Distance between parallel plates = 2cm = 0.02m
Potential difference = 100V - Find data:
Electric field strength = E =? - Formula:
- [math] E = \frac{V}{d}[/math]
- Solution
- [math] E = \frac{V}{d} \\ E = \frac{100}{0.02} \\ E = 5000vm^{-1} [/math]
4. Work done:
- The work done (W) in moving a charge (Q) through a potential difference (ΔV) against an electric field (F).
- Derivation:
- 1. Work done (W) = Force (F) * distance (d)
- [math] W = Fd \qquad (1)[/math]
-
2. Force (F)= Electric field strength (E)* Charge (Q)
- [math] F = EQ \qquad (2) [/math]
-
3. Electric field strength (E) = Potential difference (ΔV) / Distance (d)
- [math] E = \frac{ \Delta V}{d} \qquad (3)[/math]
-
Substitute equations (3) into equation (2):
- [math] F = (\frac{ \Delta V}{d}) Q \qquad (4)[/math]
- Substitute equation (4) into equation (1)
- [math] W = (\left( \frac{\Delta V}{d} \right) Q) d \\
W = Q \Delta V [/math] - So, the work done (W) is equal to the charge (Q) times the potential difference (ΔV).
- This derivation shows that the work done in moving a charge between the plates is proportional to the charge and the potential difference, which is a fundamental concept in electromagnetism!
5. Trajectory of moving charged particle entering a uniform electric field initially at right angles.
- When a charged particle enters a uniform electric field at right angles (90°), its trajectory follows a parabolic path. Here’s a step-by-step explanation:
Figure 5 A positive charge particle enter into a uniform electric field- Initial velocity (v₀): The particle enters the electric field with an initial velocity perpendicular to the field lines.
- Electric force (F): The electric field exerts a force on the particle, given by
F = qE
Where q is the charge and E is the electric field strength.
- Acceleration (a): The force causes the particle to accelerate in the direction of the electric field, with acceleration.
a = F/ q
- Parabolic trajectory: As the particle accelerates, its trajectory becomes parabolic, with the particle moving in a curved path.
- Uniform electric field: Since the electric field is uniform, the acceleration remains constant, resulting in a constant curvature of the trajectory.
- Final velocity (v): The particle’s final velocity is the vector sum of its initial velocity and the velocity gained due to acceleration.
- Note: The particle’s trajectory is a parabola because the acceleration is constant, resulting in a quadratic relationship between distance and time.
6. Radial electric field:
- Figure 6 shows a photograph of the shape of an electric field close to a small point charge. The electric field has a symmetrical radial shape near to a small point charge.
Figure 6 A small point charge move in a radical symmetrical- Figure 7 shows how we can represent the electric field lines close to a positively charged sphere.
- The lines point outwards symmetrically from the sphere as if they had come from the centre of the sphere. You can also see that the lines spread out.
- This means that the field gets weaker as the distance increases from the sphere.
- This is very similar to the shape of the gravitational field near to a planet, except that the gravitational field lines must always point towards the planet.
Figure 7 The electric field lines close to a positively charged sphere- We can produce a formula for the electric field close to a sphere as follows.
- We know from Coulomb’s law that the force between a sphere, carrying charge Q, and a small charge q at a distance r from the centre is
- [math] F = \frac{Qq}{4 \pi \varepsilon_0 r^2} [/math]
- We also know that
- [math] F = Eq [/math]
- It follows that the electric field close to the sphere is given by the formula
- [math] E = \frac{Q}{4 \pi \varepsilon_0 r^2} [/math]
- So, the strength of the electric field obeys an inverse square law.
As you can see from Figure 7, the electric field is a vector quantity.
So, when we consider the field close to two or more-point charges, we must take account of the direction of the electric field.