Ch 26: Potential and Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 26: Potential and Field

Problem 78e

Chapter 26, Problem 78e

An electric dipole at the origin consists of two charges ±q spaced distance s apart along the y-axis. What is the field Ē on the bisecting axis? Does your result agree with Equation 23.11?

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Define the electric dipole: An electric dipole consists of two charges, +q and -q, separated by a distance s. The bisecting axis is the x-axis, which is perpendicular to the line joining the charges and passes through the midpoint of the dipole.

Write the expression for the electric field due to a point charge: The electric field at a distance r from a point charge q is given by $ E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} $, where $ \epsilon_0 $ is the permittivity of free space.

Determine the contributions of the two charges to the electric field on the bisecting axis: The distance from each charge to a point on the x-axis is $ r = \sqrt{x^2 + (s/2)^2} $. The electric field due to each charge has both x and y components. The y-components cancel out because the charges are opposite in sign, leaving only the x-components.

Calculate the x-component of the electric field: The x-component of the electric field due to each charge is $ E_x = E \cos\theta $, where $ \cos\theta = \frac{x}{\sqrt{x^2 + (s/2)^2}} $. Combine the contributions from both charges to find the net electric field along the x-axis.

Simplify the expression for the net electric field: After combining the contributions and simplifying, the result should match Equation 23.11, which states that the electric field on the bisecting axis of a dipole is $ E = \frac{1}{4\pi\epsilon_0} \frac{2qxs}{(x^2 + (s/2)^2)^{3/2}} $. Verify that your derived expression agrees with this result.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a distance. It is characterized by its dipole moment, which is a vector quantity pointing from the negative to the positive charge. The electric field generated by a dipole varies with distance and direction, and it is crucial for understanding how dipoles interact with external electric fields.

Electric Field

The electric field (E) is a vector field that represents the force per unit charge experienced by a positive test charge placed in the field. For a dipole, the electric field can be calculated at various points in space, including the bisecting axis, where the field's behavior is influenced by the orientation and distance from the dipole. Understanding how to derive the electric field from a dipole is essential for solving related problems.

Equation 23.11

Equation 23.11 likely refers to a specific formula in the context of electric dipoles, typically relating to the electric field produced by a dipole at a certain point in space. This equation provides a mathematical framework for calculating the electric field based on the dipole moment and the distance from the dipole. Comparing the derived field on the bisecting axis with this equation helps validate the theoretical understanding of dipole fields.

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Textbook Question

Consider a uniformly charged sphere of radius R and total charge Q. The electric field E_out outside the sphere (r≥R) is simply that of a point charge Q. In Chapter 24, we used Gauss’s law to find that the electric field E_in inside the sphere (r≤R) is radially outward with field strength $E i n = \frac{1}{4 π ϵ_{0}} \frac{Q}{R^{3}} r$ . What is the ratio V_center/V_surface?

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Textbook Question

Derive Equation 26.33 for the induced surface charge density on the dielectric in a capacitor.

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Textbook Question

An electric dipole at the origin consists of two charges ±q spaced distance s apart along the y-axis. Assuming s≪x and y, find expressions for Ex and Ey, the components of Ē for a dipole.

Textbook Question

Charge is uniformly distributed with charge density ρ inside a very long cylinder of radius R. Find the potential difference between the surface and the axis of the cylinder.

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Textbook Question

Find an expression for the capacitance of a spherical capacitor, consisting of concentric spherical shells of radii R₁ (inner shell) and R₂ (outer shell).

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Textbook Question

An electric dipole at the origin consists of two charges ±q spaced distance s apart along the y-axis. Find an expression for the potential V(x, y) at an arbitrary point in the xy-plane. Your answer will be in terms of q, s, x, and y.

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