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Chapter 8: Problem 17

8.17 Compare and contrast (a) the properties of E-fields due to charges and B-fields due to steady currents, (b) the properties of electric charges and magnetic poles, (c) the properties of electric and magnetic dipoles.

Short Answer

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Question: Compare and contrast the properties of (a) electric fields (E-fields) due to charges and magnetic fields (B-fields) due to steady currents, (b) electric charges and magnetic poles, and (c) electric and magnetic dipoles. Answer: (a) E-fields are generated by electric charges, and their field lines start from positive charges and end at negative charges, whereas B-fields are generated by steady currents, and their field lines form closed loops. E-field strength depends on the charge and distance from the charge; B-field strength depends on the current and distance from the wire. E-fields cause forces on charged particles, while B-fields cause forces on moving charged particles. (b) Electric charges have two types (positive and negative), can exist independently, and their interaction follows the rule that like charges repel, and opposite charges attract. Magnetic poles also have two types (north and south), but they cannot exist independently, and their interaction follows the rule that like poles repel, and opposite poles attract. (c) Both electric and magnetic dipoles have dipole moments and interact with external fields, experiencing a torque that tends to align them with the field. Electric dipoles are formed by a pair of equal and opposite charges separated by a distance, while magnetic dipoles are formed by a closed current loop or a small bar magnet. Electric field lines start from the positive charge and end at the negative charge, while magnetic field lines form closed loops starting from the north pole and returning to the south pole.

Step by step solution

01

(a) Comparing E-fields and B-fields

Electric fields are created by electric charges, while magnetic fields are created by moving charges (currents). Their properties can be compared as follows: 1. Source: Electric fields are generated by electric charges, while magnetic fields are generated by steady currents (moving charges). 2. Field lines: Lines of electric fields start from positive charges and end at negative charges. In contrast, magnetic field lines are closed loops that start from the north pole of a magnet and return to the south pole. 3. Strength: The strength of an electric field is proportional to the quantity of charge and inversely proportional to the square of the distance from the charge. The strength of a magnetic field is proportional to the current and inversely proportional to the distance from the wire carrying the current. 4. Interaction: Electric fields cause forces on charged particles, whereas magnetic fields cause forces on moving charged particles. 5. Work: Electric fields do work on charged particles if they move along the field lines, whereas magnetic fields do no work on charged particles.
02

(b) Comparing electric charges and magnetic poles

Electric charges and magnetic poles are both fundamental properties of matter. Their differences and similarities can be described as follows: 1. Types: Electric charges come in two types: positive and negative. Magnetic poles come in two types as well: north and south. 2. Existence: Electric charges can exist independently, meaning that isolated positive or negative charges can be found. However, magnetic poles cannot exist independently; there are no known isolated magnetic monopoles. 3. Interaction: Like charges repel, and opposite charges attract, while like poles repel, and opposite poles attract. 4. Quantization: Electric charge is quantized, meaning it always comes in integer multiples of the elementary charge (e). It is an open question whether magnetic charge (if it exists) is quantized.
03

(c) Comparing electric and magnetic dipoles

Both electric and magnetic dipoles are characterized by their dipole moments. Their similarities and differences include: 1. Formation: An electric dipole is formed by a pair of equal and opposite electric charges separated by a distance. A magnetic dipole is formed by a closed current loop or a small bar magnet. 2. Dipole moment: The electric dipole moment is a vector pointing from the negative charge to the positive charge, with the magnitude equal to the product of the charge and the distance between them. The magnetic dipole moment is a vector in the direction of the magnetic field generated by the current loop, with magnitude proportional to the current and the area enclosed by the loop. 3. Field lines: Electric field lines of a dipole start from the positive charge and end at the negative charge. Magnetic field lines of a dipole form closed loops, starting from the north pole and returning to the south pole. 4. Interaction with fields: Electric dipoles interact with external electric fields, experiencing a torque that tends to align them with the field. Similarly, magnetic dipoles interact with external magnetic fields, experiencing a torque that tends to align them with the field. 5. Energy: Both electric and magnetic dipoles have potential energy when placed in external fields, which depends on the angle between the dipole moment and the field direction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

E-fields vs B-fields
When considering the universe of electromagnetism, electric (E-fields) and magnetic fields (B-fields) play central roles, and understanding their differences is crucial.

E-fields emerge from static electric charges, with their lines of force extending from positive to negative charges. The strength of an E-field is determined by the charge that produces it and decreases with the square of the distance from the charge. This inverse square law is a cornerstone of electric field behavior.

In contrast, B-fields originate from moving electric charges, or electric currents. Remarkably, their lines create closed loops, extending from a magnet's north pole and curving back to the south pole. This phenomenon demonstrates a key difference: while E-fields can do work moving charged particles along their lines, B-fields do not perform work, exerting force only on charged particles in motion across the field's path.

Furthermore, the B-field's strength depends on the current's magnitude and inversely on the distance from the current-carrying wire, showcasing how motion and magnetism intertwine.
Electric charges and magnetic poles
Diving into the elemental properties of matter, we find electric charges and magnetic poles, each with distinct attributes and behaviors.

Electric charges exist as two types: positive and negative. These charges can be solitary players, capable of standing alone. They are also quantized, meaning they occur in fixed, indivisible amounts—essentially, the building blocks of electrical interaction.

Meanwhile, magnetic poles appear as north and south pairs. Unlike electric charges, they are inseparable, always occurring in tandem—the elusive magnetic monopole is yet to be discovered, or it may not exist at all. Both electric charges and magnetic poles feature attractive and repulsive interactions, but while electric charges can be isolated, magnetic poles are eternally paired, and this inseparability maintains the integrity of magnetic field lines as closed loops.
Electric and magnetic dipoles
Electric and magnetic dipoles are systems with two equal but opposite types of properties separated by a distance, having both interesting similarities and differences.

An electric dipole consists of two charges of opposite sign, poised across a space. Its dipole moment—a measure of the system's separation and magnitude of charge—points from negative to positive. Consequently, electric field lines created by a dipole connect its two charges, creating a pattern that speaks volumes about the nature of the electric forces at play.

In the realm of magnetism, a magnetic dipole is akin to a tiny bar magnet or a loop of current. The magnetic dipole moment is proportional to the current and the loop size, and, unlike electric field lines, the magnetic field lines complete a continuous path. Both types of dipoles interact with their respective fields, seeking an alignment that minimizes potential energy, thus aligning the dipole moment with the field direction.
Field lines
Field lines are a visual aid crucial to understanding both electric and magnetic fields, as they illustrate the direction and strength of these fields in space.

For electric fields, the lines begin at positive charges and terminate at negative charges. The density of these lines represents the field's strength; a greater concentration of lines correlates with stronger fields. Therefore, when close to a charge, where lines are crowded, the strength is greatest.

In the magnetic universe, field lines form closed circles, stemming from north poles and looping back to south poles. This continuity of field lines reflects the non-existence of magnetic monopoles, as well as the unbroken nature of magnetic forces. These lines also never cross, preserving the uniqueness of the field's direction at any given point.
Dipole moments
The dipole moment is a vector quantity that encapsulates the essential characteristics of both electric and magnetic dipoles. It serves as an influential factor in predicting the behavior of dipoles within external fields.

For an electric dipole, the dipole moment points from the negative to the positive charge, its magnitude reflecting the product of the charge value and the separation distance. This moment is key in determining how an electric dipole will align in an external electric field, thereby affecting the force and torque it experiences.

Similar principles apply to magnetic dipoles. Their dipole moments arise from current loops or small magnets, directing along the axis from south to north pole. The strength of the magnetic dipole moment correlates with how much current flows and the area it encloses. These dipole moments not only dictate alignment in magnetic fields but also establish the potential energy and stability of the system. Through these moments, we achieve a deeper understanding of electromagnetic phenomena.

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Most popular questions from this chapter

Show that the vector potential at a point a distance \(R\) from the centre of a finite straight wire of length \(2 L\) and carrying a current \(I\) is \(\left(\mu_{0} I / 2 \pi\right) \log _{a}\left[p+\left(1+p^{2}\right)^{1 / 2}\right]\), where \(p=L / R\). Hence show that \(B\) tends to \(\mu_{0} I /(2 \pi R)\) as \(L\) tends to infinity, even though \(A\) itself tends to infinity.

An infinite solenoid has its axis along z. Using Ampère's circuital law, Gauss's law and symmetry, show that for the B-field due to a current \(I\) in the solenoid (a) \(B\), is everywhere zero, (b) \(B\), is everywhere zero, (c) \(B_{z}\) is zero outside the solenoid and equal to \(\mu_{0} n l\) everywhere inside, \(n\) being the number of turns per unit length.

A straight current-carrying wire of infinite length and negligible section lies along 2 . Show that the cylindrical components of \(B\) denoted by \(B\), and \(B_{z}\) are evetywhere zero.

A magnetic dipole of fixed moment \(m\) is moved slowly along the axis of a circular coil of radius a carrying a fixed current. How does the force on the dipole vary with its distance \(x\) from the centre of the coil if \(m\) always lies along the axis? At what distance, if any, has the force a maximum value?

A uniform magnetic field \(\mathrm{B}_{0}\) has only a \(z\)-component. Show that a vector potential given by \(A_{x}=-\frac{1}{2} B_{0} y, A_{y}=\frac{1}{2} B_{0} x, A_{n}=0\) is suitable, but that so is \(A_{x}=-B_{0} y, A,=A_{z}=0 .\) Explain.
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