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University Physics with Modern Physics

Hugh D. Young, Roger A. Freedman

Physics

14th edition Edition 路 1596 pages

ISBN: 9780321973610

Chapter 4: Q44E (page 779)

In a certain region of space the electric potential is given by $V = {Ax}^{2} y - {Bxy}^{2}$ , where $A = 5.00 V / m^{3}$ and $B = 8.00 V / m^{3}$ . Calculate the magnitude and direction of the electric field at the point in the region that has coordinates x = 2.00 m, y = 0.400 m, and z = 0.

Short Answer

Expert verified

$E = - 6.72 i - 7.2 j N / C E = 9.85 N / C 胃 = 227^{鈭�}$

Step by step solution

01

parameters.

$V = {Ax}^{2} y - {Bxy}^{2} A = 5.00 V / m^{3} B = 8.00 V / m^{3} x = 2.00 m y = 0.4 m$

02

Calculating the magnitude and direction of the electric field.

The electric field is given by

$E = - 鈭� v = E_{x} $ + E_{y} $$

The electric field is equal to the negative gradient of the potential so

$E_{x} = - \frac{鈭� V}{鈭� x} = - 2 A x y + B y^{2}$

Also in y direction

$E_{y} = - \frac{鈭� V}{鈭� y} = - A x^{2} + 2 B x y$

The electric field is given by

$E = [- 2 A x y + B y^{2}] i^+ [- A x^{2} + 2 B x y] j^$

The electric field E(2.0,0.4,0) is given by

localid="1664280297072" $E (2.0, 0.4, 0) = [(- 2) (5) (2) (0.4) + (8) {(0.4)}^{2}] i $ + [- (5) {(2)}^{2} + 2 (8) (0.4) (2)] = - 6.72 i $ - 7.2 j $ N I C$

The magnitude E I given by

$E = \sqrt{(7.2)^{2} + (6.72)^{2})} = 9.85 N / C$

Since both components are negative the angle lies in the 3rd quadrant so add to magnitude of angle

$胃 = 180 + \arctan \frac{13.6}{/ 6.72} = 227^{鈭�}$

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Most popular questions from this chapter

Questions: When a thunderstorm is approaching, sailors at sea sometimes observe a phenomenon called 鈥淪t. Elmo鈥檚 fire,鈥� a bluish flickering light at the tips of masts. What causes this? Why does it occur at the tips of masts? Why is the effect most pronounced when the masts are wet? (Hint: Seawater is a good conductor of electricity.)

A cylindrical rod has resistance $R$ . If we triple its length and diameter, what is its resistance in terms of R

A $1.50 - m$ cylindrical rod of diameter $0.500 cm$ is connected to

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reads $17.2 A$ . You can ignore any thermal expansion of the rod. Find (a) the

resistivity at and (b) the temperature coefficient of resistivity at for the material of the rod.

Current passes through a solution of sodium chloride. In

$1.00 s, 2.68 脳 10^{16} {Na}^{+}$ ions arrive at the negative electrode and $3.92 脳 10^{16} {CI}^{-}$

ions arrive at the positive electrode. (a) What is the current passing between

the electrodes? (b) What is the direction of the current?

A particle with charge $- 5.60 nC$ is moving in a uniform magnetic fieldrole="math" localid="1655717557369" $B = - (1.25 T) \hat{k}$

The magnetic force on the particle is measured to berole="math" localid="1655717706597" $\overset{鈬赌}{F} = - (3.40 脳 10^{- 7} N) \hat{i} - (7.40 脳 10^{- 7} N) \hat{j}$ (a) Calculate all the components of the velocity of the particle that you can from this information. (b) Are there
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