17.5 RELATION BETWEEN E AND V
17.5.1 Relation between F and U
Consider a positive point charge q, placed at a point A in an electric field produced by a positively charged object, as shown in Figure 17.39.
 |
| Figure 17.39 A positive point charge q, placed at a point A in an electric field produced by a positively charged object |
A repulsive electric force $F_e$ acts on q, pointing in the direction shown. Suppose qqq moves through a small distance $\Delta x$ in the same direction as the force. Then the work done on q by the electric force $F_e$ is
$\Delta W=+F_e\Delta x$
Because it moves in a direction which is the same as the direction of the electric force, the charge passes through an electric potential drop. Let the electric potential drop be of magnitude ΔU. Then we have
$\Delta W=+F_e\Delta x=-\Delta U$
or
$+F_e\Delta x=-\Delta U$
$F_e=-\frac{\Delta U}{\Delta X}$
If we let $\Delta x \to 0$, then we will get
$F_e=-\frac{dU}{dx}$
17.5.2 Relation between E and V
1. Relation between E and V
The electric force $F_x$ acting on point charge q at point A along the x-axis is given by
$F_x=qE_x$
where $E_x$ is the component of the electric field strength along the x-axis at A.
The electric potential energy U possessed by q at A is given by
$U=qV$
where V is the electric potential at A.
We have
$F_e=-\frac{dU}{dx}$
where $dU/dx$ is the rate of change of U along the x-axis.
$qE_x=-\frac{dqV}{dx}$
$E_x=-\frac{dV}{dx}$
where
$dV/dx$ is the rate of change of V along the x-axis. It is known as the electric potential gradient.
2. Unit of E
The conventional unit for E is $Vm^{-1}$. The other unit is $NC^{-1}$
3. V-x graph
The graph in Figure 17.40 shows an example of how V varies along the x-axis.
 |
| Figure 17.40 The graph shows an example of how V varies along the x-axis. |
The gradient at a point on the curve represents the electric field strength at that point and points along the x-axis.
EXAMPLE 17.24
A uniform electric field exists in a region and has a magnitude of 5.0 kV/m. Its point horizontally to the right. Three points, A, B, and C are located in the field, as shown in Figure 17.41. Determine the potential difference (a) $V_B-V_A$ and (b) $V_C-V_B$
 |
| Figure 17.41 The graph shows an three points, A, B, and C are located in the field |
Answer
(a) Since the field is uniform, the potential gradient dV/dx becomes $\Delta V/\Delta x$
$E_{A \ to \ B} =-\frac{\Delta V_{AB}}{\Delta x}$
$E_{A \ to \ B} =-\left(\frac{V_B-V_A}{AB}\right)$
$5.0 \times 10^3 =-\left(\frac{V_B-V_A}{4}\right)$
$V_B-V_A=-(5.0 \times 10^3)(4)=-20 \ kV$
This means that $V_B>V_A$.
 |
| Figure 17.42 omponent of E in the direction of CB is $E \ cos \ \theta$ |
(b) Component of E in the direction of CB is $E \ cos \ \theta$, as shown in Figure 17.42.
$E \ cos \ \theta =-\left(\frac{V_B-V_C}{CB}\right)$
$(5.0 \times 10^3) \left(\frac{4}{5}\right) =\frac{V_C-V_B}{5}$
This means that $V_B<V_C$.
17.5.3 Electric Field between a Pair of Charged Parallel Plates
 |
| Figure 17.43 Consider a pair of metal plates $P_1$ and $P_2$ placed parallel to each other, separated by distance d |
1. Electric field pattern and direction
Consider a pair of metal plates $P_1$ and $P_2$ placed parallel to each other, separated by distance d, as shown in Figure 17.43(a). They are charged by connecting a battery of emf $V_0$ to them. Then an electric field exists in the space between the plates. The field
(a) is not uniform in the space near the edges of the plates, and is represented by curved field lines
(b) is uniform elsewhere inside the space between the plates, and is represented by field lines which are straight, evenly spaced and parallel to one another.
The electric field lines point from the positively charged plate $P_1$ towards the negatively charged plate $P_2$ (which sometimes might be earthed).
2. Electric field strength
At any point in the electric field where the field is uniform, the field strength E is given by
where dV/dx represents the rate of drop in electric potential as we start moving from the positive plate $P_1$ along a line perpendicular to the plates towards plate $P_2$.
$\frac{dV}{dx} = \frac{0 - V_0}{d}$
The magnitude of the electric field strength E at any point in the uniform field is given by
If we are inside the uniform field and move from $P_1$, we will experience a uniform drop in potential per unit distance travelled from $P_1$ to $P_2$. This uniform drop in potential is shown as a straight line with a negative gradient in the graph of V against x.
$\text{gradient of line} = -\frac{V_0}{d} =E$
EXAMPLE 17.25
An oil drop carries two excess electrons. It is at rest inside the space between a pair of horizontal parallel oppositely charged plates. The p.d. across the plates is 3.0 kV and the plates are separated by 5.0 cm. Determine the mass of the oil drop. Neglect upthrust due to displacement of air by the oil drop.(Electronic charge = $-1.6 \times 10^{-19}\ C$, $g = 9.8 \ m \ s^{-2}$.
Answer
Because the oil drop is in static equilibrium, we have
weight of oil drop = electric force acting upwards on the oil drop
$mg = qE = (2e)\frac{V_0}{d}$
$m = \left(\frac{2e}{g}\right)\left(\frac{V_0}{d}\right)$
$m= \frac{(2 \times 1.6 \times 10^{-19})}{9.8} \left(\frac{3000}{0.050}\right)$
$m= 2.0 \times 10^{-15} \ kg$
17.5.5 Conclusion – Relation Between Electric Field and Electric Potential
The electric field and electric potential are closely related concepts that describe how electric forces act in space. Electric potential represents the electric potential energy per unit charge, while the electric field represents the force experienced by a charge in that region.
The electric field is directly related to the rate at which electric potential changes with distance. This relationship is expressed by
$E = -\frac{dV}{dx}$
First-order ODE visualization coming soon.
The negative sign indicates that the electric field always points in the direction of decreasing electric potential.
In a uniform electric field, such as the field between two parallel plates, the electric field strength can also be determined from the potential difference and the distance between the plates:
$E = \frac{V}{d}$
This relationship is important because it allows us to analyze electric forces, energy changes, and the motion of charged particles in electric fields. Understanding how electric field and electric potential are connected is essential for studying many electrical systems, including capacitors, electronic devices, and particle motion in electric fields.
17.5.4 Frequently Asked Questions (FAQ)-Relation Between Electric Field and Electric Potential
1. What is the relationship between electric field and electric potential?
The electric field is the rate of change of electric potential with distance. Mathematically,
First-order ODE visualization coming soon.
The negative sign shows that the electric field points in the direction of decreasing potential.
2. What does the negative sign in $E = -\frac{dV}{dx}$ mean?
It means the electric field always points from higher electric potential to lower electric potential.
3. What is electric potential difference?
Electric potential difference is the change in electric potential between two points in an electric field. It represents the work done per unit charge to move a charge between those points.
4. How does electric potential change in a uniform electric field?
In a uniform electric field, the potential decreases linearly with distance along the direction of the field.
5. What is the formula relating electric field and potential difference in a uniform field?
where
E = electric field strength, V = potential difference, d = distance between two points
5. What happens to electric potential along the direction of the electric field?
Electric potential decreases as we move along the direction of the electric field.
6. Where is electric potential highest in parallel plates?
The electric potential is highest at the positively charged plate and lowest at the negatively charged plate.
7. Can the electric field exist where potential is constant?
No. If the potential is constant (no change with distance), the electric field is zero.
8. What is an equipotential surface?
An equipotential surface is a surface where all points have the same electric potential, so no work is required to move a charge along it.
9. Why is the relation between electric field and potential important?
It helps us understand how electric forces, energy, and motion of charges behave in electric fields, which is essential in circuits, capacitors, and many electronic devices.
17.5.6 Applications – Relation Between Electric Field and Electric Potential
The relationship between electric field and electric potential is very useful in understanding many physical systems and technologies.
1. Parallel Plate Capacitors
In a parallel plate capacitor, a uniform electric field exists between the plates. The electric field strength depends on the potential difference applied across the plates and the distance between them.
This principle is used in energy storage devices, electronic circuits, and power supplies.
2. Determining Electric Field from Potential Graphs
By measuring how electric potential changes with distance, we can determine the electric field in a region. The electric field equals the negative gradient of the potential.
First-order ODE visualization coming soon.
This method is widely used in electric field mapping experiments.
3. Designing Electrical Equipment
Engineers use the relationship between electric field and potential to design capacitors, insulators, and high-voltage equipment to ensure safe and efficient operation.
4. Motion of Charged Particles
Charged particles move because of the electric field created by potential differences. This principle is applied in devices such as electron tubes, cathode ray tubes, and particle accelerators.
Electrostatic precipitators, photocopiers, and laser printers use electric fields created by potential differences to control and move charged particles.
6. Electric Field Visualization
The relation between electric potential and electric field helps scientists understand the shape and direction of electric fields, which is important in electrostatics and electromagnetic studies.
Post a Comment for "Relation Between Electric Field and Electric Potential (E = −dV/dx) – Derivation, Formula and Graph"