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Force Fields
“Action at a distance” (across “empty” space)—how is this possible?
This happens with gravitational force and it happens with electrical
force. Newton wrestled with this question and never resolved it—
because he thought only of a material connection (i.e. some bit of
matter) acting as the agent of the force. But obviously force and
energy can be transmitted across a complete vacuum (outer space)!
This led physicists over the 150 years after Newton to accept the idea
that space itself has properties.
A field is a point-by-point description of some property of space
itself. A force field, for example, describes the magnitude and
direction of the force that will be exerted on a body when it is placed
at a given point in that space.
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Even when considering force fields, the parallels remain between
gravitational and electrostatic forces:
The gravitational force, FG, on a body of mass, m, located at a point P,
is given by FG = mg, where g is the gravitational force field at point P:
g = –GmP/r122 r^
In other words, g is the property of that point P in space; and m is the
property of the body placed there. Notice that FG and g are vectors
here; they have magnitudes and directions. If we know m and g, we
can compute FG. Or, if we know m and FG, we can find g. …
Example: A brick of mass 3.0 kg weighs 6.0 N on the moon’s surface.
Find the magnitude and direction of the gravitational field on the lunar
surface (expressed in x-y axes).
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Likewise, the electrostatic force, FE, on a body with net charge, q,
located at a point P, is given by FE = qE, where E is the electric force
field—at point P. In other words, E is the property of that point P in
space; and q is the property of the body placed there.
Notice the units of E.…
FE and E are vectors—with magnitudes and directions. If we know q
and E, we can compute FE; if we know q and FE, we can find E.…
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Example: An electrostatic force of 6.0 N is acting on a point charge of
3.0 C. Calculate the magnitude and direction of the electric field, E,
that must be present at the location of the charge.
Again, the analogy:
To find the gravitational force, FG, on an object, we must know its
mass, m, and the gravitational force field, g, at its location.
To find the electrical force, FG, on an object, we must know its charge,
q, and the electrical force field, E, at its location.
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The Force and the Field
Coulomb’s Law gives the magnitude of the electrostatic force between
two point charges: |FE| = k|q||q0|/r2.
But we now know we can also express FE as the product of the charge
that feels that force and a property of a point in space—the vector
quantity that we call the electric field at that point. For example, the
force magnitude felt by q0 (exerted by q) would be |FE| = q0|E|.
Thus, k|q||q0|/r2 = q0|E|, or in other words: |E| = k|q|/r2
This is the magnitude of the electric field, E, that is caused by q and
felt by q0.
Note: A charge does not feel the E-field it causes (just as a mass does
not feel its own gravity).
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Notice: E-fields help us use the signs of the charges to indicate the
directions of the forces they feel.
E-field lines begin at (point away from) a positive point charge. Efield lines terminate (point toward) a negative point charge.
Using FE = q0E (full vector notation now), we can easily decide the
direction of the force felt by any charge q0 placed into that field:
If a positive charge, q0, is placed into an E-field, it feels a force in the
same direction as the E-field lines at that point. If a negative charge,
q0, is placed into an E-field, it feels a force in the direction opposite
to the E-field lines at that point.
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More examples:
What is the strength and direction of the electric field 1 nm from
a proton?
What force would an electron feel if placed in the field of that
proton 1 nm away?
(See After Class 3 for full solutions and discussions.)
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Notice, too: The total E-field at a given point is the vector sum of the
fields caused by individual point charges.
This principle of superposition is fundamental to understanding
fields and using them conveniently.
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All three charges shown here
have equal magnitude and are
equidistant from each other.
At the position of the dot,
the electric field points…
A.
B.
C.
D.
E.
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Up the page.
Down the page.
To the left.
To the right.
The electric field is zero.
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All three charges shown here
have equal magnitude and are
equidistant from each other.
At the position of the dot,
the electric field points…
A.
B.
C.
D.
E.
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Up the page.
Down the page.
To the left.
To the right.
The electric field is zero.
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Consider the four field patterns. Which pattern(s) could
represent a field caused by one or a few fixed point charges?
1.
2.
3.
4.
5.
6.
(a)
(b)
(b) and (d)
(a) and (c)
(b) and (c)
Some other combination
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Consider the four field patterns. Which pattern(s) could
represent a field caused by one or a few fixed point charges?
1.
2.
3.
4.
5.
6.
(a)
(b)
(b) and (d)
(a) and (c)
(b) and (c)
Some other combination
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Examples:
Point charge A (qA = +2e) sits at x = 0.
Point charge B (qB = +8e) sits at x = 3.
Find the total E-field magnitude produced by the
charges at the point x = 2 (all distances in µm).
(How about at x = 1?)
(See After Class 3 for full solutions and discussions.)
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