In a region of space where there are no conduction or...

In a region of space where there are no conduction or displacement currents, it is impossible to have a uniform magnetic field that abruptly drops to zero. To prove this statement, use the method of contradiction: Assume that such a case is possible, and then show that your assumption contradicts a law of nature. (a) In the bottom half of a piece of paper, draw evenly spaced horizontal lines representing a uniform magnetic field to your right. Use dashed lines to draw a rectangle abcda with horizontal side $a b$ in the magnetic-leld region and horizontal side $c d$ in the top half of your paper where $B=0 .$ (b) Show that integration around your rectangle contradicts Ampere's law, Eq. $(29.15) .$

Key Concepts

Proof by Contradiction

Proof by contradiction is a logical reasoning method where one assumes the opposite of what is to be proven and then shows that this assumption leads to a contradiction with a known law or fact. In the context of magnetic fields, assuming the existence of a uniform field that abruptly drops to zero in a current-free region leads to a contradiction of Ampere’s Law, thereby demonstrating that such a scenario is physically impossible.

Closed Loop Integration

Closed loop integration is a mathematical technique often used in electromagnetism to apply integral forms of Maxwell’s equations, such as Ampere’s Law. By evaluating the line integral of the magnetic field around a closed contour, one can infer properties about the current distribution or, in regions devoid of currents, verify that the net contribution must vanish. This provides a powerful tool to analyze the behavior of fields and to show inconsistencies when assumed field configurations do not meet physical law constraints.

Ampere's Law

Ampere's Law is one of the fundamental equations in electromagnetism which relates the circulation of a magnetic field around a closed loop to the total electric current (and in Maxwell’s extension, the displacement current) passing through the surface bounded by that loop. In regions where there are no conduction or displacement currents, the line integral of the magnetic field around any closed path should be zero, thus enforcing continuity in the field and preventing abrupt changes.

Magnetic Field Continuity

In electromagnetic theory, the continuity of the magnetic field implies that its value and direction cannot change abruptly in space if no source, such as a current or discontinuity in material properties, is present. This principle follows from Maxwell’s equations and ensures that in inert regions (without currents or charges), magnetic fields must vary smoothly, precluding sudden jumps from a finite value to zero.

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