Fig. 3 shows an electric dipole in a uniform electric field with magnitude 10 × 105

Question

Fig. 3 shows an electric dipole in a uniform electric field with magnitude 10 × 105

N/C directed parallel to the plane of the figure. The electric dipole consists of two
charges 𝑞1 = +2𝑒 and 𝑞2 = −2𝑒 ( 𝑒 = 1.6 × 10−19), which both lie in the plane and
separated by a distance 0.13𝑛𝑚.
3.1 Find
3.1.1 the net force exerted by the field on the dipole
3.1.2 the magnitude and direction of the electric dipole moment
3.1.3 the magnitude and direction of the Torque
3.1.4 the potential energy of the system in position shown in Fig. 3.

Answer 1

To find the net force exerted by the field on the dipole, we need to consider the electric field experienced by each charge in the dipole.

The electric field experienced by charge q1 is given by:
E1 = k * (q1 / r1^2)
where k is the electrostatic constant (9 x 10^9 Nm^2/C^2), q1 is the charge of q1 (+2e = 2 * 1.6 x 10^-19 C), and r1 is the distance from q1 to the center of the dipole.

The electric field experienced by charge q2 is given by:
E2 = k * (q2 / r2^2)
where q2 is the charge of q2 (-2e = -2 * 1.6 x 10^-19 C) and r2 is the distance from q2 to the center of the dipole.

Since the electric field is uniform, the magnitudes of E1 and E2 are equal.

Using the formula for the net force on a dipole in an electric field, we have:
F = Δp * E
where Δp is the electric dipole moment and E is the electric field.

The electric dipole moment (p) is given by:
p = q * d
where q is the magnitude of either charge (2e = 2 * 1.6 x 10^-19 C) and d is the separation between the charges (0.13 nm = 0.13 x 10^-9 m).

To find the magnitude and direction of the torque, we use the formula:
τ = p * E * sin(θ)
where θ is the angle between the dipole moment and the electric field.

Finally, the potential energy of the system is given by:
U = -p * E * cos(θ)

Now let's calculate these values:

3.1.1 The net force exerted by the field on the dipole:
First, we need to calculate the magnitudes of E1 and E2:

E1 = k * (q1 / r1^2) = (9 x 10^9 Nm^2/C^2) * (2e / (0.13 x 10^-9 m)^2)
E1 = 9 x 10^9 Nm^2/C^2 * (2 * 1.6 x 10^-19 C) / (0.13 x 10^-9 m)^2

Similarly,
E2 = k * (q2 / r2^2) = (9 x 10^9 Nm^2/C^2) * (-2e / (0.13 x 10^-9 m)^2)
E2 = 9 x 10^9 Nm^2/C^2 * (-2 * 1.6 x 10^-19 C) / (0.13 x 10^-9 m)^2

Since the electric field is uniform, the net force is the product of Δp and E:

F = Δp * E
F = (q * d) * (E1 - E2)

3.1.2 The magnitude and direction of the electric dipole moment:
p = q * d

3.1.3 The magnitude and direction of the torque:
τ = p * E * sin(θ)

3.1.4 The potential energy of the system:
U = -p * E * cos(θ)

Note: In order to fully solve the problem and provide specific numerical calculations, the values of q, d, r1, and r2 are needed.

Answer 2

To solve these questions, we can use the formulas related to electric dipole in a uniform electric field.

3.1.1 The net force exerted by the field on the dipole can be found using the formula:

𝐹 = 𝑝𝐸

where 𝐹 is the net force, 𝑝 is the electric dipole moment, and 𝐸 is the electric field.

3.1.2 The magnitude and direction of the electric dipole moment can be found using the formula:

𝑝 = 𝑞 × 𝑑

where 𝑞 is the charge and 𝑑 is the distance between the charges.

3.1.3 The magnitude and direction of the torque can be found using the formula:

𝜏 = 𝑝 × 𝐸 × 𝑠𝑖𝑛(𝜃)

where 𝜃 is the angle between the dipole moment vector and the electric field vector.

3.1.4 The potential energy of the system can be found using the formula:

𝑈 = −𝑝 × 𝐸 × 𝑑

Let's substitute the given values into these formulas:

𝑞1 = +2𝑒 = +2 × 1.6 × 10^(-19) C = +3.2 × 10^(-19) C
𝑞2 = −2𝑒 = −2 × 1.6 × 10^(-19) C = −3.2 × 10^(-19) C
𝑑 = 0.13 nm = 0.13 × 10^(-9) m
𝐸 = 10 × 10^5 N/C (parallel to the dipole moment)

Let's calculate each part step by step:

3.1.1 The net force exerted by the field on the dipole:

𝐹 = 𝑝𝐸
= (𝑞1 + 𝑞2)𝐸
= (+3.2 × 10^(-19) C + (-3.2 × 10^(-19) C)) × (10 × 10^5 N/C)

Solving this expression will give the net force.

3.1.2 The magnitude and direction of the electric dipole moment:

𝑝 = 𝑞 × 𝑑
= (+3.2 × 10^(-19) C) × (0.13 × 10^(-9) m)

Solving this expression will give the magnitude of the electric dipole moment. The direction will be from the negative charge to the positive charge.

3.1.3 The magnitude and direction of the torque:

𝜏 = 𝑝 × 𝐸 × 𝑠𝑖𝑛(𝜃)

In this case, we need the angle between the dipole moment and the electric field vector. Since the field is parallel to the dipole moment, the angle is 0 degrees. Thus, sin(𝜃) = sin(0) = 0.

The torque will be zero.

3.1.4 The potential energy of the system:

𝑈 = −𝑝 × 𝐸 × 𝑑
= −(+3.2 × 10^(-19) C × (10 × 10^5 N/C) × (0.13 × 10^(-9) m)

Solving this expression will give the potential energy of the system.