Chapter 1

Electricity and Magnetism

Dr. Mahmoud Wahdan

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The Electric Field

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The Electric Field (Cont.)

At any point, the field is taken to be in the direction of the force an object would experience there.

The strength of the field is proportional to the strength of that force.

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The Electric Field (Cont.) Fig. 1.1: the gravitational field of the earth

– If an object is placed at point A, it will experience a force in the direction of the arrowhead, toward the earth’s center.

– The lines, called field lines, show the direction of the earth’s gravitational pull; this is taken to be the direction of the gravitational field.

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Fig 1.1: The gravitational field of the earth is directed radially inward and becomes stronger as one approaches the earth.

The Electric Field (Cont.) Field lines not only represent the direction

of the force but also Indicate its relative magnitude.

In Fig. 1.1, we notice that the lines are closer together near the earth, where the force is strong, than they are farther away from the earth, where the force is weaker.

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The Electric Field (Cont.) The electric field represents the electric

force a stationary positive charge experiences.

How you determine the electric field in a region?– Placing a charged object (call it a test object) in

the region and determine the force on it due to all other charges.

– Test charge exerts forces on all other charges in the vicinity and, if these charges are in metals, could cause them to move.

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The Electric Field (Cont.) We imagine that the test charge has a very

special property: the test charge is a fictitious charge that exerts no forces on nearby charges.

Test charge qt

For example, suppose a positive test charge is placed at point A in Fig. 1.2a.

It is attracted radially, as shown by the arrow at A.

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The Electric Field (Cont.)The force on the positive test charge is directed

radially inward no matter where in the neighborhood of the central negative charge it is placed.

The electric field is directed as shown by the arrows: the electric field near a negative charge is directed radially into the charge.

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Fig 1.2: The electric field is directed radially inward toward a negative charge and radially outward from a positive charge.

The Electric Field (Cont.) In Fig. 1.2b:

The positive test charge is repelled radially outward by the central positive charge.

The electric field near a positive charge is directed radially away from the charge.

The directed lines in Fig. 1.2 show the direction of the electric field.

These directed lines are called electric field lines.

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The Electric Field (Cont.) Electric field lines originate on and are

directed away from the positive charges, and terminate on and are directed toward negative charges.

Definition: electric field strength E At any given point, the direction of E, a vector

quantity, is taken to be the same as that of the electric field line through that point.

The magnitude of E is equal to the force experienced by the test charge divided by the amount of charge qt:

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The Electric Field (Cont.)E = F / qt (N/C)

Definition: electric field (E)

The electric field at a point in space is defined as the ratio of the electric force a small positive test charge qt experiences at that point to the magnitude of the test charge

E = F (on qt) / qt

Where:

1. The direction of E is the same as the direction of the force F on a positive charge.

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The Electric Field (Cont.)2. The IS units of E are N/C.

3. A corollary of the definition of E is that the force on a charge q placed at a point where the electric field has the value E is

F = q E

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The Electric Field of a Point Charge

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Suppose we wish to find the electric field strength at point P in Fig.1.3, which is a distance r away from a positive charge q.

The electric field due to q is redially outward (Fig. 1.2b).

E at point P is in the direction shown (Fig. 1.3).

Fig 1.3: To find the electric field E at point P, we must compute the force a positive test charge would experience if placed at that point.

The Electric Field of a Point Charge (Cont.)

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If a test charge qt placed at P, the force on it is given by Coulomb's law:

F = k q qt / r 2

Dividing by qt to obtain F/qt, the electric field strength

F/qt = k q / r 2

E = k q / r 2 for a point charge (1.1)

The Electric Field of a Point Charge (Cont.)

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Where:

1.The direction of the electric field is radially outward from a positive charge, radially toward a negative charge.

2.The electric field due to a number of point charges can in principle be calculated at any point by applying the superposition principle: calculate the field due to each point charge separately and then add the individual contributions vectorially.

3.In a map of the electric field, the strength of the field is greatest where the field lines are densest and least where the lines are farthest apart.

The Electric Field of a Point Charge (Cont.)

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Illustration 1.1

Find the electric field strength 50 cm from a positive point charge of 1 x 10-4C.

Reasoning

Finding E at point P in Fig. 1.3 with r = 0.50 m and q = 1 x 10-4C.

Because q is positive, the test charge placed at P is repelled outward by q.

The direction of E is as shown.

The Electric Field of a Point Charge (Cont.)

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The magnitude of E (use Eq. 1.1)

E = k q / r 2

= (9 x 109 N.m2/C2) 1 X 10-4 C / (0.50 m)2

= 3.6 x 106 N/C

The Electric Field of a Point Charge (Cont.)

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Example 1.1

Find the magnitude of E at point B in Fig. 1.4 due to the two point charges.

Fig 1.4: Justify the directions shown for E1 and E2. How do we find the total field at B resulting from the two charges?

The Electric Field of a Point Charge (Cont.)

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The Electric Field of a Point Charge (Cont.)

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The Electric Field of a Point Charge (Cont.)

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Illustration 1.2

If a charge q = +4 x 10-7C were placed at point B in Example 1.1, what force would be exerted on it by the electric field?

Reasoning

Using Coulomb’s law and calculate the force as in Example 1.1.

Once we have calculated the field E at point B, the force any charge q experiences when placed at that point is just F = q E.

The Electric Field of a Point Charge (Cont.)

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The magnitude of the force:

F = (+4 x 10-7 C) (7.3 x 106 N/C) = 2.9 N

The direction of the force is the direction of qE. In the case of a positive charge, F is in the direction of E. In the case of a negative charge, F is in the direction of –E, or opposite E.