Problem 3:
An infinite plane in the xz plane carries a uniform surface charge density
1 = 15 nC/m2.
A second infinite plane carrying a uniform charge density 2 = 10 nC/m2 intersects the
x-y plane at the z-axis and makes an angle of 30° with the xz plane
as shown in the figure. Find the electric field in the xy plane at
This is problem 19-42 in Tipler, one of the list of "suggested extra
problems".
It is really just a vector addition problem (remember electric field
is a vector). The field from each plane can be calculated
separately and then added together as vectors (this is the
`superposition principle'). The field from an infinite plane is given
by E = 2
k (see front cover of the test), is perpendicular to
the surface, and directed away from the surface (for positive charge
density). Thus we just need to add two vectors, one from each surface.
The only complication is: are we above both, below both, or between
the surfaces?
In part a) the point (6,2) is clearly between the two surfaces (the arctan of (2/6) is 18° which is less than 30°, but still above the xz plane). Thus the field from the first surface is just
E1 = 2k1 j
and from the second surface is
E2 = 2k2(sin 30° i -
cos 30° j)
Adding these vectors by adding their components, and plugging in the values of
1, 2. and k = 9 x 109 Nm2/C2 we get
E = E1 + E2 = 282 N/C i + 358 N/C j
In part b) the location is clearly above the 30° plane, so the field from this one is reversed in direction; otherwise everything else is the same and we get
E1 = 2k1 j
E2 = 2k2(-sin 30° i +
cos 30° j)
E = 1337 N/C j - 282 N/C i
Problem 4
Test 2
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