Lecture 2 - Professor Li Gao PDF

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PHYS 220B ELECTRICITY AND MAGNETISM

Lecture #2 Chapter 21: Electric Charge and Electric Field 21.4 Electric Field and Electric Forces The electric force on a charged body is exerted by the electric field created by other charged bodies. A charged body produces an electric field in the space around it. We use a small test charge 𝑞0 to find out if an electric field is present.

Electric field is defined as electric force per unit charge. 󰇍𝑬 =

󰇍0 𝑭 𝑞0

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PHYS 220B ELECTRICITY AND MAGNETISM

Electric field of a point charge: - The location of the charge is called the source point. - The point P where we are determining the field is called the field point.

The vector equation for the force 󰇍𝑭0 :

󰇍0= 𝑭

1 𝑞𝑞0 𝒓 4𝜋𝜖0 𝑟 2

The vector equation for the electric field at P: 󰇍 = 𝑬

1 𝑞 𝒓 4𝜋𝜖0 𝑟 2

𝐹0 =

1 |𝑞𝑞0 | 4𝜋𝜖0 𝑟 2

The magnitude of the force:

The magnitude of the electric field at P: 𝐸=

1 |𝑞| 4𝜋𝜖0 𝑟 2

Example 21.6: A point charge 𝑞 = −8.0 nC is located at the origin. Find the electric-field vector at the field point 𝑥 = 1.2 m, 𝑦 = −1.6 m. 󰇍𝑬 = (−11ii󰆹 + 14jj󰆹) N/C

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PHYS 220B ELECTRICITY AND MAGNETISM

21.5 Electric Field Calculations Calculate electric fields caused by various distributions of electric charge. 1. Electric field of an electric dipole: Example 21.8

2. Electric field of a ring of charge: Example 21.9

-

Assume the charge is positive. Due to the symmetry, the net electric field at P is along the x axis.

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PHYS 220B ELECTRICITY AND MAGNETISM

󰇍𝑬 = 𝐸𝑥 i󰆹 = -

𝑄𝑥 1 󰆹i 4𝜋𝜖0 (𝑥 2 + 𝑎 2 )3/2

Consider two extreme conditions: 𝑥 = 0 𝑎𝑛𝑑 𝑥 ≫ 𝑎

3. Electric field of a charged line segment: Example 21.10

-

-

Find 𝑑𝐸𝑥 and 𝑑𝐸𝑦 and integrate.

󰇍 = 𝑬

1 4𝜋𝜖0

𝑥(𝑥 2

󰇍 depends on the sign of 𝑄. The direction of 󰇍𝑬 4/9

𝑄

+

󰆹

1i 𝑎 2 )2

PHYS 220B ELECTRICITY AND MAGNETISM

-

Discuss two extreme conditions: (1) 𝑥 ≫ 𝑎 (2) 𝑥 ≪ 𝑎

󰇍𝑬󰇍 = 󰇍 = 𝑬

1 2𝜋𝜖0

1 𝑄 i󰆹 4𝜋𝜖0 𝑥 2 𝜆

𝑥 2 𝑥 √( ) + 1 𝑎

𝜆 󰆹 i i󰆹 = 2𝜋𝜖0 𝑥

4. Electric field of a uniformly charged disk: Example 21.11

-

Represent the charge distribution as a collection of concentric rings of charge 𝑑𝑄 . 𝐸𝑥 =

-

1 𝜎 [1 − ] 2𝜖0 √(𝑅 2 /𝑥 2 ) + 1

Consider one extreme condition: 𝑥 ≪ 𝑅

𝐸𝑥 =

𝜎 2𝜖0

So the electric field produced by an infinite plane sheet of charge is independent of the distance from the sheet. The field direction is everywhere perpendicular to the sheet, away from it (for positive charge). If the surface charge density is negative, the field directions on both sides of the plane are toward it. 5. Electric field of two oppositely charged infinite sheets: Example 21.12

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PHYS 220B ELECTRICITY AND MAGNETISM

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PHYS 220B ELECTRICITY AND MAGNETISM

Questions: 1. Two large, flat, horizontally oriented plates are parallel to each other, a distance d apart. Half way between the two plates the electric field has magnitude E. If the separation of the plates is reduced to d/2 what is the magnitude of the electric field half way between the plates? A) 4E B) 2E C) E D) 0 E) E/2 Answer: C 2. The figure shows two unequal point charges, q and Q, of opposite sign. Charge Q has greater magnitude than charge q. In which of the regions X, Y, Z will there be a point at which the net electric field due to these two charges is zero?

A) only regions X and Z B) only region X C) only region Y D) only region Z E) all three regions Answer: B 3. An electron is initially moving to the right when it enters a uniform electric field directed upwards. Which trajectory shown below will the electron follow?

A) trajectory W B) trajectory X C) trajectory Y D) trajectory Z Answer: D 4. A very long wire carries a uniform linear charge density of 7.0 nC/m. What is the electric field strength 16.0 m from the center of the wire at a point on the wire's perpendicular 2 -12 2 bisector? (ε0 = 8.85 × 10 C /N ∙ m ) 7/9

PHYS 220B ELECTRICITY AND MAGNETISM

A) 7.9 N/C B) 3.9 N/C C) 0.49 N/C D) 0.031 N/C Answer: A 5. Three point charges lie at the vertices of an equilateral triangle as shown. All three charges have the same magnitude, but charges #1 and #2 are positive (+q) and charge #3 is negative (–q). The net electric force that charges #2 and #3 exert on charge #1 is in A) the +x direction B) the –x direction C) the +y direction D) the –y direction E) none of the above Answer: D

6. Two point charges and a point P lie at the vertices of an equilateral triangle as shown. Both point charges have the same magnitude q but opposite signs. There is nothing at point P. The net electric field that charges #1 and #2 produce at point P is in A) the +x direction B) the –x direction C) the +y direction D) the –y direction E) none of the above Answer: C

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PHYS 220B ELECTRICITY AND MAGNETISM

7. Positive charge is uniformly distributed around a semicircle. The electric field that this charge produces at the center of curvature P is in A) the +x direction B) the –x direction C) the +y direction D) the –y direction E) none of the above Answer: D

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