Question

If V = 0 at a point in space, must E ?= 0? If E ?= 0 at some
point, must V = 0 at that point?

Explain. Give examples for each.

b. Can two equipotential lines cross? Explain.

Answer #1

A)

The charge of V over a very short distance is E, they are not
explicitly dependent quantities. So both answer is no. For example
In the center of a dipole V = 0 but E is finite. And the point
midway between two positive charges. **E** is 0 there,
but *V* is finite.

B)

The electric field on an equipotential line is the perpendicular on the tangent at that point. If, say, two equipotential lines cross then at the same point electric fields will have two different directions which is absurd. So two equipotential lines can't cross.

Choose the correct statement: A. If E = 0 at a point P then V
must be negative at P B. The potential of a negatively charged
conductor must be positive C. If V = 0 at a point P then E must be
positive at P D. None of the above are correct E. An electron tends
to go from a region of high potential to a region of low
potential

Did any of your field lines cross? Should they? Why, or why
not?
None of the field lines crossed, nor should they. They are a
line of constant potential and the equipotential can have only one
value at a given point in space. If the electric field lines were
to intersect, they would render a location with two different
strong electric field vectors and so would not accurately represent
equipotential lines.
Did any of your equipotential surfaces cross? Should they?...

The electric potential due to a point charge is given by
V = k q / r where q is the charge, r is the
distance from q, and k =8.99×109Nm2⁄C2.
A) Show, in detail, that the SI unit of
electric potential is a volt.
B) What are equipotential lines?
C) How are equipotential lines used to obtain
the electric field lines?

An electron travels with v?
=5.40×106i^m/sthrough a point in space where E?
=(1.70×105i^?1.70×105j^)V/m and B?
=?0.140k^T.
What is the force on the electron?

Calculate numerical values for V and ρν at point P in free space
if: (a)V = 4yz x2 +1 , atP(1,2,3); (b) V = 5ρ2 cos2φ, atP(ρ = 3,φ =
π 3 , z =2); (c) V = 2cosφ r2 , atP(r =0.5,θ=45◦, φ =60◦). Ans. 12
V,−106.2 pC/m3;−22.5 V, 0; 4 V, 0

Suppose that, in some vector space V , a vector u ∈V has the
property that u+v = v
for some v ∈ V . Prove that u = 0.

A point in space has a voltage of 4.5 V. Will a charged particle
accelerate at that location? Explain how you know – or if you don’t
know, explain what additional information would help you determine
the answer.

Equipotential Surface Plotting a) How much work is done by the
electrostatic force on a point charge that is moved on an
equipotential surface (line)? Explain. b) If the electric field
lines are not normal to the equipotential surfaces, what would
happen?

The electric potential (V) at a certain point in space is given
by: V(x, y, z) = 5x2-3xy+xyz
a) find the directional derivative of the potential at P(3,4,5)
in the direction of the vector v=i+j+k
b) calculate the gradient of the electric potential

Select True or False for the following statements about electric
field lines.
E-field lines point outward from positive charges.
E-field lines point inward toward negative charges.
E-field lines may cross.
A positive point charge released from rest will initially
accelerate along an E-field line.
E-field lines make circles around positive charges.
E-field lines do not begin or end in a charge-free region except at
infinity.
Where the E-field lines are dense the E-field must be weak.

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