Physics
Electric Field of Multiple Point Charges
The electric field of multiple point charges is the combined effect of the individual electric fields produced by each charge. It is determined by vector addition of the electric fields due to each charge at a given point in space. The resulting electric field at that point is the vector sum of the individual electric fields.
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8 Key excerpts on "Electric Field of Multiple Point Charges"
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
The electric field lines for such a situation are shown in Fig. 22.1.4. We also get Eq. 22.5.6 if we let z → 0 in Eq. 22.5.5 while keeping R finite. This shows that at points very close to the disk, the electric field set up by the disk is the same as if the disk were infinite in extent. Checkpoint 22.5.1 In Fig. 22.5.1, as we sweep the ring outward, what happens to the ring’s contribution of electric field at point P? 22.6 A POINT CHARGE IN AN ELECTRIC FIELD Learning Objectives After reading this module, you should be able to . . . 22.6.1 For a charged particle placed in an external electric field (a field due to other charged objects), apply the relationship between the electric field E → at that point, the particle’s charge q, and the electrostatic force F → that acts on the particle, and identify the relative directions of the force and the field when the particle is positively charged and negatively charged. 22.6.2 Explain Millikan’s procedure of measuring the elementary charge. 22.6.3 Explain the general mechanism of ink-jet printing. 681 22.6 A POINT CHARGE IN AN ELECTRIC FIELD Figure 22.6.1 The Millikan oil-drop apparatus for measuring the elemen- tary charge e. When a charged oil drop drifted into chamber C through the hole in plate P 1 , its motion could be controlled by closing and opening switch S and thereby setting up or eliminating an electric field in cham- ber C. The microscope was used to view the drop, to permit timing of its motion. Insulating chamber wall + – B S P 2 C Oil drop P 1 A Microscope Oil spray A Point Charge in an Electric Field In the preceding four modules we worked at the first of our two tasks: given a charge distribution, to find the electric field it produces in the surrounding space. Here we begin the second task: to determine what happens to a charged particle when it is in an electric field set up by other stationary or slowly moving charges.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
630 CHAPTER 22 Electric Fields Our goals in this chapter are to (1) define electric field, (2) discuss how to calculate it for various arrangements of charged particles and objects, and (3) discuss how an electric field can affect a charged particle (as in making it move). The Electric Field A lot of different fields are used in science and engineering. For example, a tem- perature field for an auditorium is the distribution of temperatures we would find by measuring the temperature at many points within the auditorium. Similarly, we could define a pressure field in a swimming pool. Such fields are examples of scalar fields because temperature and pressure are scalar quantities, having only magnitudes and not directions. In contrast, an electric field is a vector field because it is responsible for conveying the information for a force, which involves both magnitude and direc- tion. This field consists of a distribution of electric field vectors E → , one for each point in the space around a charged object. In principle, we can define E → at some point near the charged object, such as point P in Fig. 22.1.2a, with this proce- dure: At P, we place a particle with a small positive charge q 0 , called a test charge because we use it to test the field. (We want the charge to be small so that it does not disturb the object’s charge distribution.) We then measure the electrostatic force F → that acts on the test charge. The electric field at that point is then E → = F → ___ q 0 (electric field). (22.1.1) Because the test charge is positive, the two vectors in Eq. 22.1.1 are in the same direction, so the direction of E → is the direction we measure for F → . The mag- nitude of E → at point P is F/q 0 . As shown in Fig. 22.1.2b, we always represent an electric field with an arrow with its tail anchored on the point where the mea- surement is made.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
Definition of the Electric Field The electric field E B that exists at a point is the electrostatic force F B experienced by a small test charge* q 0 placed at that point divided by the charge itself: E B 5 F B q 0 (18.2) The electric field is a vector, and its direction is the same as the direction of the force F B on a positive test charge. SI Unit of Electric Field: newton per coulomb (N/C) Equation 18.2 indicates that the unit for the electric field is that of force divided by charge, which is a newton/coulomb (N/C) in SI units. It is the surrounding charges that create an electric field at a given point. Any pos- itive or negative charge placed at the point interacts with the field and, as a result, experi- ences a force, as the next example indicates. *As long as the test charge is small enough that it does not disturb the surrounding charges, it may be either positive or negative. Compared to a positive test charge, a negative test charge of the same magnitude experiences a force of the same magnitude that points in the opposite direction. However, the same electric field is given by Equation 18.2, in which F B is replaced by 2 F B and q 0 is replaced by 2q 0 . EXAMPLE 7 | An Electric Field Leads to a Force In Figure 18.15 the charges on the two metal spheres and the ebonite rod create an electric field E B at the spot indicated. This field has a magnitude of 2.0 N/C and is directed as in the drawing. Determine the force on a charge placed at that spot, if the charge has a value of (a) q 0 5 118 3 10 28 C and (b) q 0 5 224 3 10 28 C. Reasoning The electric field at a given spot can exert a variety of forces, depending on the magnitude and sign of the charge placed there. The charge is assumed to be small enough that it does not alter the locations of the surrounding charges that create the field.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Our goals in this chapter are to (1) define electric field, (2) discuss how to calculate it for various arrangements of charged particles and objects, and (3) dis- cuss how an electric field can affect a charged particle (as in making it move). The Electric Field A lot of different fields are used in science and engineering. For example, a tem- perature field for an auditorium is the distribution of temperatures we would find by measuring the temperature at many points within the auditorium. Similarly, we could define a pressure field in a swimming pool. Such fields are examples of scalar fields because temperature and pressure are scalar quantities, having only magnitudes and not directions. In contrast, an electric field is a vector field because it is responsible for conveying the information for a force, which involves both magnitude and direc- tion. This field consists of a distribution of electric field vectors E → , one for each point in the space around a charged object. In principle, we can define E → at some point near the charged object, such as point P in Fig. 22-2a, with this proce- dure: At P, we place a particle with a small positive charge q 0 , called a test charge because we use it to test the field. (We want the charge to be small so that it does not disturb the object’s charge distribution.) We then measure the electrostatic force F → that acts on the test charge. The electric field at that point is then E → = F → q 0 (electric field). (22-1) Because the test charge is positive, the two vectors in Eq. 22-1 are in the same direction, so the direction of E → is the direction we measure for F → . The magnitude of E → at point P is F/q 0 . As shown in Fig. 22-2b, we always represent an electric field with an arrow with its tail anchored on the point where the measurement is made.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
DEFINITION OF THE ELECTRIC FIELD The electric field E → that exists at a point is the electrostatic force F → experienced by a small test charge* q 0 placed at that point divided by the charge itself: E → = F → q 0 (18.2) The electric field is a vector, and its direction is the same as the direction of the force F → on a positive test charge. SI Unit of Electric Field: newton per coulomb (N/C) Equation 18.2 indicates that the unit for the electric field is that of force divided by charge, which is a newton/coulomb (N/C) in SI units. It is the surrounding charges that create an electric field at a given point. Any positive or negative charge placed at the point interacts with the field and, as a result, experiences a force, as the next example indicates. EXAMPLE 7 An Electric Field Leads to a Force In Figure 18.16 the charges on the two metal spheres and the ebonite rod create an electric field E → at the spot indicated. This field has a magnitude of 2.0 N/C and is directed as in the drawing. Determine the force on a charge placed at that spot, if the charge has a value of (a) q 0 = +18 × 10 ‒8 C and (b) q 0 = ‒24 × 10 ‒8 C. Reasoning The electric field at a given spot can exert a variety of forces, depending on the magnitude and sign of the charge placed there. The charge is assumed to be small enough that it does not alter the loca- tions of the surrounding charges that create the field. Solution (a) The magnitude of the force is the product of the magnitudes of q 0 and E → : F = |q 0 | E = (18 × 10 −8 C)(2.0 N/C) = 36 × 10 −8 N (18.2) Since q 0 is positive, the force points in the same direction as the electric field, as part a of the drawing indicates. (b) In this case, the magnitude of the force is F = |q 0 | E = (24 × 10 −8 C)(2.0 N/C) = 48 × 10 −8 N (18.2) The force on the negative charge points in the direction opposite to the force on the positive charge—that is, opposite to the electric field (see part b of the drawing).
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
Also, as you did with the gravitational field of an object with mass, you should picture the electric field of a charge-bearing object (the source charge) as a continuous, immaterial substance that surrounds the source charge, filling all of space—in principle, to ±∞ in all directions. The field exists at every physical point in space. To put it another way, the electric charge on an object alters the space around the charged object in such a way that all other electrically charged objects in space experience an electric force as a result of being in that field. The electric field, then, is the mechanism by which the electric properties of the source charge are transmitted to and through the rest of the universe. (Again, the range of the electric force is infinite.) We will see in subsequent chapters that the speed at which electrical phenomena travel is the same as the speed of light. There is a deep connection between the electric field and light. Superposition Yet another experimental fact about the field is that it obeys the superposition principle. In this context, that means that we can (in principle) calculate the total electric field of many source charges by calculating the electric field of only q 1 at position P, then calculate the field of q 2 at P, while—and this is the crucial idea—ignoring the field of, and indeed even the existence of, q 1 . We can repeat this process, calculating the field of each individual source charge, independently of the existence of any of the other charges. The total electric field, then, is the vector sum of all these fields. That, in essence, is what Equation 5.4 says. In the next section, we describe how to determine the shape of an electric field of a source charge distribution and how to sketch it. The Direction of the Field Equation 5.4 enables us to determine the magnitude of the electric field, but we need the direction also.
- David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
• If more than one charged particle sets up an electric feld at a point, the net electric feld is the vector sum of the individual electric felds — electric felds obey the superposition principle. To find the electric field due to a charged particle (often called a point charge), we place a positive test charge at any point near the particle, at distance r. From Coulomb’s law, the force on the test charge due to the particle with charge q is F = 1 4 0 qq 0 r 2 r. As previously, the direction of F is directly away from the particle if q is positive (because q 0 is positive) and directly towards it if q is negative. We can now write the electric field set up by the particle (at the location of the test charge) as E = F q 0 = 1 4 0 q r 2 r. (22.2) If given another charged particle, we can immediately determine the directions of the electric field vectors near it by just looking at the sign of the charge q. We can find the magnitude at any given distance r by converting the vector equation to a magnitude form: E = 1 4 0 |q| r 2 = k |q| r 2 . (22.3) We write |q| to avoid the danger of getting a negative E when q is negative and then thinking that the negative sign has something to do with direction. This last equation gives magnitude E only. We must think about the direction separately. Figure 22.10 gives a number of electric field vectors at points around a positively charged particle. Pdf_Folio:480 480 Fundamentals of physics Be careful with such figures. Each vector represents the vector quantity at the point where the tail of the arrow is anchored. The vector is not something that stretches from a ‘here’ to a ‘there’ as with a displacement vector. FIGURE 22.10 The electric feld vectors at various points around a positive point charge.
- Luis Manuel Braga de Costa Campos(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
561 8 Electric/Magnetic Multipoles and Images A line charge in space corresponds to the generalized function two-dimensional unit impulse, and the corresponding influence or Green function for the Poisson equation specifies the potential due to the line charge. This method is distinct from, and leads to the same results as, the theory of the complex potential (Chapters I.12, 24, and 26), not only for monopoles, but also for dipoles, quadru-poles, and multiple expansions. Whereas the method using complex functions is restricted to plane problems, the use of generalized functions is valid regardless of spatial dimension; for example, the latter method extends (Section 9.2) readily from two (Chapter I.11) to three (Chapter 8) or higher (Sections 9.7 through 9.9) dimensions. The three-dimensional unit impulse represents a point charge, and the associated influence or Green function specifies the corresponding potential; the convolution integral expressing the principle of superposition then specifies the potential due to an arbitrary distribution of charges. This leads to the three-dimensional laws for the Coulombian force field which are related to plane laws (Section I.24.4), but distinct (Section 8.3) from them. In space (Section 8.1), as in the plane (Section I.12.7), a dipole is the limit of two opposite monopoles whose strength increases inversely with distance; whereas the monopole is specified by a unit impulse and is radially symmetric, a dipole involves the gradient of a unit impulse projected in the direction of the vector dipole moment and, thus, is axisymmetric (Section 8.2). A quadrupole is (Section 8.3) the limit of opposite dipoles and involves second-order derivatives of the unit impulse; since it has two axes, it is no longer generally axisymmetric, and the quadrupole moment is represented by a matrix.
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