24. Electric Force & Field; Gauss' Law

A 1.5μC charge, with a mass of 50g, is in the presence of an electric field that perfectly balances its gravity. What magnitude does the electric field need to be, and in what direction does it need to point?

If two equal charges are separated by some distance, they form an electric dipole. Find the electric field at the center of an electric dipole, given by the point P in the following figure, formed by a 1C and a −1C charge separated by 1 cm.

4 charges are arranged as shown in the following figure. Find the magnitude of the electric field at the center of the arrangement, indicated by the point P.

In the following figure, a mass m is balanced such that its tether is perfectly horizontal. If the mass is m and the angle of the electric field is θ, what is the magnitude of the electric field, E, expressed in terms of m, q, and θ?

(II) An electron moving to the right at 7.5 x 10⁵ m/s enters a uniform electric field parallel to its direction of motion. If the electron is to be brought to rest in the space of 5.0 cm,

(a) what direction is required for the electric field, and

(b) what is the strength of the field?

(I) A proton is released in a uniform electric field, and it experiences an electric force of 1.68 x 10⁻¹⁴ N toward the south. Find the magnitude and direction of the electric field.

(II) Determine the magnitude of the acceleration experienced by an electron in an electric field of 756 N/C. How does the direction of the acceleration depend on the direction of the field at that point?

(II) What is the electric field strength at a point in space where a proton experiences an acceleration of 2.4 million “g’s”?

(II) Calculate the electric field at the center of a square 42.5 cm on a side if one corner is occupied by a ―33.8 μC charge and the other three are occupied by ―22.0 μC charges.

(II) At what position, 𝓍 = 𝓍ₘ , is the magnitude of the electric field along the axis of the ring of Example 21–10 a maximum?

(II) Estimate the electric field at a point 2.40 cm perpendicular to the midpoint of a uniformly charged 2.00-m-long thin wire carrying a total charge of 7.45 μC.

(II) Two point charges, Q₁ = ― 32 μC and Q₂ = +45μC , are separated by a distance of 12 cm. The electric field at the point P (see Fig. 21–61) is zero. How far from Q₁ is P? <IMAGE>

(II) You are given two unknown point charges, Q₁ and Q₂. At a point on the line joining them, one-third of the way from Q₁ to Q₂, the electric field is zero (Fig. 21–64). What is the ratio Q₁/Q₂? <IMAGE>

(II) A positive charge q is placed at the center of a circular ring of radius R. The ring carries a uniformly distributed negative charge of total magnitude ― Q.

(a) If the charge q is displaced from the center a small distance x as shown in Fig. 21–71, show that it will undergo simple harmonic motion when released.

(b) If its mass is m, what is its period?

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A point charge of mass 0.145 kg, and net charge +3.40 μC, hangs at rest at the end of an insulating cord above a large sheet of charge. The horizontal sheet of fixed uniform charge creates a uniform vertical electric field in the vicinity of the point charge. The tension in the cord is measured to be 5.18 N.

(a) Calculate the magnitude and direction of the electric field due to the sheet of charge (Fig. 21–80).

(b) What is the surface charge density σ(C/m²) on the sheet?

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Three very large square charged planes are arranged as shown (on edge) in Fig. 21–78. From left to right, the planes have charge densities per unit area of -0.50μC/m² , +0.25 μC/m² and -0.35 μC/m². Find the total electric field (direction and magnitude) at the points A, B, C, and D. Assume the plates are much larger than the distance AD.

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(II) Determine the direction and magnitude of the electric field at the point P shown in Fig. 21–66. The two point charges are separated by a distance of 2a. Point P is on the perpendicular bisector of the line joining the charges, a distance x from the midpoint between them. Express your answers in terms of Q, x, a, and k. <IMAGE>

(II) A very long uniformly charged wire (linear charge density λ = 2.5 C/m) lies along the x axis in Fig. 21–59. A small charged sphere (Q = ―2.0 C) is at the point x = 0 cm , y = ― 5.0 cm? . What is the electric field at the point x = 7.0 cm, y = 7.0 cm? Eᵥᵥᵢᵣₑ (→ above E) and Eq (→ above E ) represent fields due to the long wire and the charge Q, respectively.

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