Problem 69 - University Physics with Modern Physics

Question

It is impossible to have a uniform electric field that abruptly drops to zero in a region of space in which the magnetic field is constant and in which there are no electric charges. 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 electric field to your right. Use dashed lines to draw a rectangle abcda with horizontal side ab in the electric-field region and horizontal side \(c d\) in the top half of your paper where \(E=0 .\) (b) Show that integration around your rectangle contradicts Faraday's law, Eq. \((29.21) .\)

Step-by-Step Solution

Answer

An abrupt change in electric field contradicts Faraday's Law because it requires a time-varying magnetic field, which is absent here.

Step 1: Visual Representation

Draw evenly spaced horizontal lines to the right in the bottom half of a sheet of paper to represent the uniform electric field (\(\mathbf{E} eq 0\)). Then draw a dashed rectangle abcda such that the side "ab" is in the field region and the side "cd" is in the top half with \(\mathbf{E}=0\). This step establishes the hypothetical scenario for our proof by contradiction.

Step 2: Apply Faraday's Law

Faraday's Law states that the line integral of the electric field \(\mathbf{E}\) around a closed loop is equal to the negative rate of change of magnetic flux through the loop: \[ \oint_{C} \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt}. \]In our hypothesis, the electric field abruptly vanishes, leading to an integral around the rectangle that includes both zero and non-zero components of \(\mathbf{E}\). This suggests a non-zero integral unless \(\frac{d\Phi_B}{dt}\) has an appropriate value.

Step 3: Evaluating the Integral

Consider the integral of \(\mathbf{E}\cdot d\mathbf{l}\) around the path abcda:- Segment "ab": \(\int_{a}^{b} \mathbf{E} \cdot d\mathbf{l} = E \times ab\).- Segment "bc" and "da": these integrals are 0 because they lie in regions where \(\mathbf{E}=0\).- Segment "cd": Back in the uniform field, \(\int_{c}^{d} \mathbf{E} \cdot d\mathbf{l} = -E \times cd\) because it is in the opposite direction.Thus, \[ \oint_{abcda} \mathbf{E} \cdot d\mathbf{l} = E(ab - cd). \] Since the field is assumed uniform, \(ab = cd\), implying\[ \oint_{abcda} \mathbf{E} \cdot d\mathbf{l} = 0. \] This means the non-zero field section must balance the integral, but \(\oint_{C} \mathbf{E} \cdot d\mathbf{l}\) should already be zero if \(\frac{d\Phi_B}{dt} = 0\).

Step 4: Derive the Contradiction

Given \(\frac{d\Phi_B}{dt} = 0\) (no time-varying magnetic field is present as per the problem), the only way for the integral to be zero is if \(E=0\) throughout or no changes at all are present around the path. However, our assumption that \(E\) could be non-zero in some parts of the path already leads to a contradiction since it violates Faraday's Law unless \(\frac{d\Phi_B}{dt}\) provides compensatory voltage, which it doesn't in this setting.

Key Concepts

Electric Field

An electric field represents the force field created around electrically charged objects, exerting force on other charges within the vicinity. Simply put, it is the region around a charged particle where an electric force is exerted on other charges. For a uniform electric field, this force is consistent in magnitude and direction at every point in the field. This characteristic makes it easy to depict with evenly spaced lines pointing directionally across the field.
In our problem, this uniform electric field is visualized in the bottom half of a paper. The field is represented by horizontal lines moving consistently to the right. Although typically, electric fields exert influences seamlessly, the hypothetical scenario challenges this as it implies an abrupt shift to no electric field, which intuitively seems puzzling.
Understanding this behavior is essential, especially when applying laws like Faraday's Law to derive conclusions about electric and magnetic fields and how they interrelate.

Magnetic Field

Magnetic fields arise from moving electric charges and are integral to understanding electromagnetism. They represent regions where a magnetic force acts on other moving charges or magnetic materials. Magnetic fields, like electric fields, have both magnitude and direction, but they are associated with the notion of magnetic flux—a measure of the field lines passing through an area.
In this context, the region of interest has a constant magnetic field while we are examining changes, or the lack thereof, in the electric field. A magnetic field generally remains unaffected unless influenced by varying currents or changes in electric fields surrounding it. Therefore, a static magnetic field, as in this scenario, indicates no dynamic processes altering the field lines' density or orientation, thereby influencing how we interpret Faraday's Law in these conditions.

Contradiction Proof

The method of contradiction is a classic approach to proving the validity or invalidity of a statement. In this exercise, we assume the possibility that a uniform electric field can suddenly drop to zero, despite it seeming counterintuitive. The goal is to highlight logical inconsistency with known laws, in this case, Faraday's Law.
By sketching a rectangle in the supposed field setup, where half the region has a non-zero electric field and the other half does not, it sets the stage for analysis. Calculating the line integral of this setup should demonstrate the inconsistency as the expected zero integral isn’t satisfied due to the abrupt change.
This contradiction leads us back to the realization that our initial assumption must be false under the conditions given, as it violates fundamental principles such as the conservation of energy.

Line Integral

In calculus, a line integral is a method to integrate a function along a curve. With electric fields, this involves the path integral of the field, assessing how much field line presence accumulates around a closed loop. In situations involving electromagnetic concepts, this is crucial for applying Faraday’s Law, which relates the electric field’s behavior with changes in magnetic flux.
In this scenario, the loop defined by the rectangle combines segments within and outside the electric field. Calculating the line integral around this loop highlights differences where \(\mathbf{E} \) is non-zero versus zero, compounded by segments where \(\mathbf{E} \) reverses direction.
Summing these parts results in a zero value for the integral if the field remains uniform, adhering to Faraday’s Law. However, in our hypothetical setup without this uniform distribution, the desired zero integral value demonstrates the impossibility of the configuration without a transformative influence—which contradicts the supposed constancy of the magnetic field and lack of time-change dynamics, thus proving the setup impossible.