Problem 32 - Physics

Question

A long, thin rod (length \(=4.0 \mathrm{~m}\) ) lies along the \(x\) axis, with its midpoint at the origin. In a vacuum, \(\mathrm{a}+8.0 \mu \mathrm{C}\) point charge is fixed to one end of the rod, and \(a-8.0 \mu C\) point charge is fixed to the other end. Everywhere in the \(x, y\) plane there is a constant external electric field (magnitude \(\left.=5.0 \times 10^{3} \mathrm{~N} / \mathrm{C}\right)\) that is perpendicular to the rod. With respect to the \(z\) axis, find the magnitude of the net torque applied to the rod.

Step-by-Step Solution

Answer

The net torque on the rod is 0.16 N·m.

Step 1: Identify the Torque Formula

To calculate the net torque due to an electric field, we use the formula \( \tau = p \times E \), where \( p \) is the dipole moment and \( E \) is the electric field. The torque will only be maximum when \( p \) is perpendicular to \( E \).

Step 2: Determine the Dipole Moment

The dipole moment \( p \) is calculated as \( p = q \cdot d \) where \( q \) is the charge at one end and \( d \) is the separation between the charges. Here, \( q = 8.0 \times 10^{-6} \) C and \( d = 4.0 \) m, so \( p = 8.0 \times 10^{-6} \times 4.0 \). Therefore, \( p = 3.2 \times 10^{-5} \; \text{C}\cdot\text{m} \).

Step 3: Calculate the Torque

The torque \( \tau \) is obtained from \( \tau = pE \) since the electric dipole is perpendicular to the electric field. Substituting the given field \( E = 5.0 \times 10^{3} \; \text{N/C} \), we get \( \tau = 3.2 \times 10^{-5} \times 5.0 \times 10^{3} \). This results in \( \tau = 0.16 \; \text{N}\cdot\text{m} \).

Key Concepts

Dipole Moment

The dipole moment is a fundamental concept in physics, especially in the study of electric fields and electrostatics. It represents a pair of equal and opposite charges separated by a distance. In simple terms, a dipole consists of two charges, typically denoted by \(+q\) and \(-q\), held apart by a distance \(d\). The dipole moment \(p\) is mathematically expressed as:

• \(p = q \cdot d\)

Here, \(q\) is the magnitude of the charge and \(d\) is the distance between the charges. The dipole moment is a vector quantity; its direction goes from the negative to the positive charge.

A unique aspect is that the dipole moment helps us understand how the dipole interacts with an external electric field. This interaction, which involves torque, is crucial in many physical and chemical phenomena. For instance, molecules with significant dipole moments may align with the electric field, a principle that has applications in understanding the behavior of molecules in external electric fields.

Electric Field

An electric field is an invisible force field that surrounds electric charges and affects other charges in the field's vicinity. It is a vector field, meaning it has both magnitude and direction. The electric field \(E\) at a point in space is defined as the electric force \(F\) experienced per unit positive charge \(q\) placed at that point, given by the formula:

• \(E = \frac{F}{q}\)

The SI unit of the electric field is newtons per coulomb (N/C).

Electric fields can arise from point charges, line charges, surface charges, and even time-varying magnetic fields. In our specific exercise, a constant electric field is applied perpendicular to the length of the rod. The strength of this field affects the dipole placed in it, exerting torque on the dipole and causing it to rotate.

Point Charge

A point charge is an idealized model of a charge that is considered to have no size, only magnitude. It simplifies the study of electrostatics by allowing us to focus on the effects of charge magnitude and location without considering the details of its geometry.

In electrostatics, the electric potential at a distance \(r\) from a point charge \(q\) is given by:

• \(V = \frac{kq}{r}\)

where \(k\) is Coulomb's constant. The electric field due to a point charge is radially outward (for positive charges) or inward (for negative charges), based on the sign of the charge.

Point charges play an essential role in understanding the behavior of electric dipoles as they provide a simplified means to analyze how accumulations of charge within objects like rods influence external electric fields and interact with other charges. In our exercise, the point charges at either end of the rod collectively form a dipole, exerting torque when placed perpendicular to the electric field, illustrating how point charges can exhibit complex interactions when combined.