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PHYS 3122, Spring 2020

Homework 1

Policy: On all homeworks, use whatever resources you need, including Griﬃths, other texts, talking to peers, oﬃce hours, etc. Remember that in the end, what you turn in must be your own work, reﬂecting your own understanding. Copying or working directly from solutions (found online, or from other students) is not considered "collaboration". Use your judgment, and remember these homeworks are intended to help you learn the material. Please show your work or explain your reasoning whenever possible.

Due Date: Wednesday 01/15 (paper: in class or under my door Howey N115; electronically: Canvas).

Problem A [20%]. Some classic results on triple products:

(a) Prove the Grassmann identity for arbitrary vectors A, B, C e R3:

A × (B × C) = B (A . C) - C (A . B)

(b) Use the Grassmann identity to derive the Jacobi identity:

A × (B × C) + B × (C × A) + C × (A × B) = 0

Problem B [30%]. A thick disk (radius R, thickness t; think of a hockey puck) of an unknown material carries a volume charge density 口 = ae −βs2 cos (e_) where a, 8 and e are constants.

(a) What is the total charge Q on the disk in terms of a, 8 and e ?

(b) What are the dimensions (physical units) of a, 8 and e ?

Problem C [30%]. A circular loop of radius r carries a uniform linear charge density o.

(a) Find the electric ﬁeld E at point P located at a distance _ above the center of the loop. Deﬁne a coordinate system adequate for the cylindrical geometry of the problem.

(b) Express the magnitude of the electric ﬁeld in the limit of _ > r (very far from the loop)? Comment on the physics of your result. [Hint: use a Taylor expansion to ﬁrst order in the small parameter ε = r/_]

(c) When approaching the loop from _ = +o, what is the sign and value of the ﬁrst correction to the above result? Comment on the physics of your result. [ Hint: (1 + ε)α ≈ 1 + aA + a(a - 1)ε2 /2]

TURN PAGE

Problem D [20%]. The book lists particularly useful double and triple product rules involving a derivative.

For gradients, important double and triple products are (f and g are scalars):

For the divergence, important double and triple products are:

For the curl, important double and triple products are:

(a) Pick two of the above rules (the choice is yours) and derive them.

(b) Now, let’s assume that A = y + 2z + 3_ and B = 3z - 2y . Pick one of rules (ii), (iv) or (vi) and check it for vectors A and B (calculating each term separately).